An Improved Implementation of the LBFGS Algorithm for Automatic History Matching
- Guohua Gao (Chevron Corp.) | Albert C. Reynolds (U. of Tulsa)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- March 2006
- Document Type
- Journal Paper
- 5 - 17
- 2006. Society of Petroleum Engineers
- 5.1.5 Geologic Modeling, 5.2.1 Phase Behavior and PVT Measurements, 5.1.2 Faults and Fracture Characterisation, 5.6.9 Production Forecasting, 4.1.5 Processing Equipment, 5.5 Reservoir Simulation, 5.5.8 History Matching, 5.6.4 Drillstem/Well Testing, 4.1.2 Separation and Treating, 4.3.4 Scale, 2.2.2 Perforating, 5.1.1 Exploration, Development, Structural Geology
- 0 in the last 30 days
- 730 since 2007
- Show more detail
- View rights & permissions
|SPE Member Price:||USD 12.00|
|SPE Non-Member Price:||USD 35.00|
For large scale history matching problems, where it is not feasible to compute individual sensitivity coefficients, the limited memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) is an efficient optimization algorithm, (Zhang and Reynolds, 2002; Zhang, 2002). However, computational experiments reveal that application of the original implementation of LBFGS may encounter the following problems: (i) converge to a model which gives an unacceptable match of production data; (ii) generate a bad search direction that either leads to false convergence or a restart with the steepest descent direction which radically reduces the convergence rate; (iii) exhibit overshooting and undershooting, i.e., converge to a vector of model parameters which contains some abnormally high or low values of model parameters which are physically unreasonable. Overshooting and undershooting can occur even though all history matching problems are formulated in a Bayesian framework with a prior model providing regularization.
We show that the rate of convergence and the robustness of the algorithm can be significantly improved by: (1) a more robust line search algorithm motivated by the theoretical result that the Wolfe conditions should be satisfied; (2) an application of a data damping procedure at early iterations or (3) enforcing constraints on the model parameters. Computational experiments also indicate that (4) a simple rescaling of model parameters prior to application of the optimization algorithm can improve the convergence properties of the algorithm although the scaling procedure used can not be theoretically validated.
Minimization of a smooth objective function is customarily done using a gradient based optimization algorithm such as the Gauss- Newton (GN) method or Levenberg-Marquardt (LM) algorithm. The standard implementations of these algorithms (Tan and Kalogerakis, 1991; Wu et al., 1999; Li et al., 2003), however, require the computation of all sensitivity coefficients in order to formulate the Hessian matrix. We are interested in history matching problems where the number of data to be matched ranges from a few hundred to several thousand and the number of reservoir variables or model parameters to be estimated or simulated ranges from a few hundred to a hundred thousand or more. For the larger problems in this range, the computer resources required to compute all sensitivity coefficients would prohibit the use of the standard Gauss- Newton and Levenberg-Marquardt algorithms. Even for the smallest problems in this range, computation of all sensitivity coefficients may not be feasible as the resulting GN and LM algorithms may require the equivalent of several hundred simulation runs. The relative computational efficiency of GN, LM, nonlinear conjugate gradient and quasi-Newton methods have been discussed in some detail by Zhang and Reynolds (2002) and Zhang (2002).
|File Size||1 MB||Number of Pages||13|
Anterion, F., Eymard, R., and Karcher, B.: "Use of Parameter Gradients forReservoir History Matching," paper SPE 18433 presented at the 1989 SPESymposium on Reservoir Simulation, Houston, 6-8 February.
Chavent, G., Dupuy, M., and Lemonnier, P.: "History Matching by Use of OptimalTheory," SPEJ (1975) 15, No. 1, 74; Trans., AIME,259.
Chen, W.H., Gavalas, G.R., Seinfeld, J.H., and Wasserman, M.L.: "A New Algorithm for AutomaticHistoryMatching," SPEJ (1974) 14, No. 6, 593; Trans., AIME,257.
Dennis, J.E. and Schnabel, R.B.: "Numerical Methods for UnconstrainedOptimization and Nonlinear Equations," 1996 SIAM, Philadelphia.
Floris, F.J.T., Bush, M.D., Cuypers, M., Roggero, F., and Syversveen, A.-R.:"Methods for quantifying the uncertainty of production forecasts: A comparativestudy," Petroleum Geoscience (2001) 7 (Supp.), 87.
Gao, G. and Reynolds, A.C.: "An improved implementation of the LBFGSalgorithm for automatic history matching with application to the PUNQ dataset," TUPREP Research Report 21 (2004) 14.
Li, R., Reynolds, A.C., and Oliver, D.S.: "History Matching of Three-Phase FlowProduction Data," SPEJ (2003) 8, No. 4, 328.
Nocedal, J.: "Updating quasi-Newton matrices with limited storage," Math.Comp. (1980) 35, No. 151, 773.
Nocedal, J. and Wright, S.J.: Numerical Optimization, Springer, NewYork City (1999).
Tan, T.B. and Kalogerakis, N.: "A Fully Implicit, Three-Dimensional,Three-Phase SimulatorWith Automatic History-Matching Capability," paper SPE21205 presented at the 1991 SPE Symposium on Reservoir Simulation, Anaheim,California, 17-20 February.
Tarantola, A.: Inverse Problem Theory: Methods for Data Fitting and ModelParameter Estimation, Elsevier, Amsterdam, The Netherlands (1987).
Wu, Z., Reynolds, A.C., and Oliver, D.S.: "Conditioning Geostatistical Models toTwo-Phase Production Data," SPEJ (1999) 4, No. 2, 142.
Zhang, F.: "Automatic History Matching of Production Data for Large ScaleProblems," PhD dissertation, U. of Tulsa, Tulsa, Oklahoma (2002).
Zhang, F. and Reynolds, A.C.: "Optimization algorithms for automatic historymatching of production data," Proc., 2002 European Conference on theMathematics of Oil Recovery, Frieberg, Germany, 3-6 September.
Zhang, F., Skjervheim, J.A., Reynolds, A.C., and Oliver, D.S.: "Automatic History Matching in aBayesian Framework, Example Applications," SPEREE (2005) 8,No. 3, 214.