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Semiautomatic Multiple Resolution Design for History Matching

Authors
Baoyan Li (Baker Hughes) | Francois Friedmann (California Institute of Technology)
DOI
https://doi.org/10.2118/102277-PA
Document ID
SPE-102277-PA
Publisher
Society of Petroleum Engineers
Source
SPE Journal
Volume
12
Issue
04
Publication Date
December 2007
Document Type
Journal Paper
Pages
408 - 419
Language
English
ISSN
1086-055X
Copyright
2007. Society of Petroleum Engineers
Disciplines
5.1.2 Faults and Fracture Characterisation, 5.5 Reservoir Simulation, 5.1.5 Geologic Modeling, 4.1.5 Processing Equipment, 2.4.3 Sand/Solids Control, 5.5.8 History Matching, 4.3.4 Scale, 2.2.2 Perforating
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Summary

History matching is an inverse problem in which an engineer calibrates key geological/fluid flow parameters by fitting a simulator's output to the real reservoir production history. It has no unique solution because of insufficient constraints. History-match solutions are obtained by searching for minima of an objective function below a preselected threshold value. Experimental design and response surface methodologies provide an efficient approach to build proxies of objective functions (OF) for history matching. The search for minima can then be easily performed on the proxies of OF as long as its accuracy is acceptable.

In this paper, we first introduce a novel experimental design methodology for semi-automatically selecting the sampling points, which are used to improve the accuracy of constructed proxies of the nonlinear OF. This method is based on derivatives of constructed proxies.
We propose an iterative procedure for history matching, applying this new design methodology. To obtain the global optima, the proxies of an objective function are initially constructed on the global parameter space. They are iteratively improved until adequate accuracy is achieved. We locate subspaces in the vicinity of the optima regions using a clustering technique to improve the accuracy of the reconstructed OF in these subspaces.

We test this novel methodology and history-matching procedure with two waterflooded reservoir models. One model is the Imperial College fault model (Tavassoli et al. 2004). It contains a large bank of simulation runs. The other is a modified version of SPE9 (Killough 1995) benchmark problem. We demonstrate the efficiency of this newly developed history-matching technique.

Introduction

History matching (Eide et al. 1994; Landa and Güyagüler 2003) is an inverse problem in which an engineer calibrates key geological/fluid flow parameters of reservoirs by fitting a reservoir simulator's output to the real reservoir production history. It has no unique solution because of insufficient constraints.

The traditional history matching is performed in a semi-empirical approach, which is based on the engineer's understanding of the field production behavior. Usually, the model parameters are adjusted using a one-factor-at-a-time approach. History matching can be very time consuming, because many simulation runs may be required for obtaining good fitting results.

Attempts have been made to automate the history-matching process by using optimal control theory (Chen et al. 1974) and gradient techniques (Gomez et al. 2001). Also, design of experiment (DOE) and response surface methodologies (Eide et al. 1994; Box and Wilson 1987; Montgomery 2001; Box and Hunter 1957; Box and Wilson 1951; Damsleth et al. 1992; Egeland et al. 1992; Friedmann et al. 2003) (RSM) were introduced in the late 1990s to guide automatic history matching. The goal of these automatic methods is to achieve reasonably faster history-matching techniques than the traditional method.

History matching is an optimization problem. The objective is to find the best of all possible sets of geological/fluid flow parameters to fit the production data of reservoirs. To assess the quality of the match, we define an OF (Atallah 1999). For history-matching problems, an objective function is usually defined as a distance (Landa and Güyagüler 2003) between a simulator's output and reservoir production data. History-matching solutions are obtained by searching for minima of the objective function. Experimental design and response surface methodologies provide an efficient approach to build up hypersurfaces (Kecman 2001) of objective functions (i.e., proxies of objective functions with a limited number of simulation runs for history matching). The search for minima can then be easily performed on these proxies as long as their accuracy is acceptable. The efficiency of this technique depends on constructing adequately accurate objective functions.

File Size  2 MBNumber of Pages   12

References

Algorithms and Theory of Computation Handbook. 1999. M.J. Atallah.New York: CRC Press LLC, 34-17.

Box, G.E.P. and Hunter, J.S. 1957. Multifactor Experimental Designs forExploring Response Surfaces. Annals of Mathematical Statistics28: 195-242.

Box, G.E.P. and Wilson, K.G. 1951.On the Experimental Attainment of OptimumConditions. Journal of Royal Statistical Society B. 13: 1-45.

Box, G.E.P. and Wilson, K.G. 1987. Empirical Model Building and ResponseSurfaces. New York: Wiley.

Chen, W.H., Gavalas, G.R., Seinfeld, J.H., and Wasserman, M.L. 1974. A New Algorithm for Automatic HistoryMatching. SPEJ 14 (6): 593-608; Trans., AIME, 257.SPE-4545-PA. DOI: 10.2118/4545-PA.

Damsleth, E., Hage, A., and Volden, R. 1992. Maximum Information at Minimum Cost:A North Sea Field Development Study With an Experimental Design. JPT44 (12): 1350-1356. SPE-23139-PA. DOI: 10.2118/23139-PA.

Egeland, T., Holden, L., and Larsen, E.A. 1992. Designing Better Decisions. PaperSPE 24275 presented at the SPE European Petroleum Computer Conference,Stavanger, 24-27 May. DOI: 10.2118/24275-MS.

Eide, A.L., Holden, L., Reiso, E., and Aanonsen, S.I. 1994. AutomaticHistory Matching by Use of Response Surfaces and Experimental Design. Paperpresented at the European Conference on the Mathematics of Oil Recovery, Røros,Norway, 7-10 June.

Friedmann, F., Chawathé, A., and Larue, D.K. 2003. Assessing Uncertainty in ChannelizedReservoir Using Experimental Designs. SPEREE 6 (4): 264-274.SPE-85117-PA. DOI: 10.2118/85117-PA.

Gomez, S., Gosselin, O., and Barker, J.W. 2001. Gradient-Based History Matching Witha Global Optimization Method. SPEJ 6 (2): 200-208.SPE-71307-PA. DOI: 10.2118/71307-PA.

Kecman, V. 2001. Learning and Soft Computing, Support Vector Machines,Neural Networks, and Fuzzy Logic Models. Cambridge, Massachusetts: The MITPress, 451.

Killough, J.E. 1995. Ninth SPEComparative Solution Project: A Examination of Black-Oil Simulation. PaperSPE 29110 presented at the SPE Reservoir Simulation Symposium, San Antonio,Texas, 12-15 February. DOI: 10.2118/29110-MS.

Landa, J.L. and Güyagüler, B. 2003. A Methodology for History Matchingand the Assessment of Uncertainties Associated With Flow Prediction. PaperSPE 84465 presented at the SPE Annual Technical Conference and Exhibition,Denver, 5-8 October. DOI: 10.2118/84465-MS.

Li, B. 2003. A Control Volume Function Approximation Method for ReservoirSimulation Using Unstructured Grids. PhD dissertation. Dallas: SouthernMethodist University.

Li, B. and Friedmann, F. 2005. Novel Multiple Resolutions Design ofExperiment/Response Surface Methodology for Uncertainty Analysis of ReservoirSimulation Forecasts. Paper SPE 92853 presented at the SPE ReservoirSimulation Symposium, The Woodlands, Texas, 31 January-2 February. DOI:10.2118/92853-MS.

Montgomery, D.C. 2001. Design and Analysis of Experiments. Fifthedition. New York: John Wiley & Sons.

Shampine, L.F., Allen, R.C., and Pruess, S. 1997. Fundamentals ofNumerical Computing. New York: John Wiley & Sons, 101-115.

Tavassoli, Z., Carter, J.N., and King, P.R. 2004. Errors in History Matching.SPEJ 9 (3): 352-361. SPE-86883-PA. DOI: 10.2118/86883-PA.

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