Abstract
It is common to formulate the history-matching problem using Bayes’ theorem. From Bayes’, the conditional probability density function (pdf) of the uncertain model parameters is proportional to the prior pdf of the model parameters, multiplied by the likelihood of the measurements. The static model parameters are random variables characterizing the reservoir model while the observations include, e.g., historical rates of oil, gas, and water produced from the wells. The reservoir prediction model is assumed perfect, and there are no errors besides those in the static parameters. However, this formulation is flawed. The historical rate data only approximately represent the real production of the reservoir and contain errors. History-matching methods usually take these errors into account in the conditioning but neglect them when forcing the simulation model by the observed rates during the historical integration. Thus, the model prediction depends on some of the same data used in the conditioning. The paper presents a formulation of Bayes’ theorem that considers the data dependency of the simulation model. In the new formulation, one must update both the poorly known model parameters and the rate-data errors. The result is an improved posterior ensemble of prediction models that better cover the observations with more substantial and realistic uncertainty. The implementation accounts correctly for correlated measurement errors and demonstrates the critical role of these correlations in reducing the update’s magnitude. The paper also shows the consistency of the subspace inversion scheme by Evensen (Ocean Dyn. 54, 539–560 2004) in the case with correlated measurement errors and demonstrates its accuracy when using a “larger” ensemble of perturbations to represent the measurement error covariance matrix.
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References
Aanonsen, S.I., Naevdal, G., Oliver, D.S., Reynolds, A., Valles, B.: Ensemble Kalman filter in reservoir engineering – a review. SPE J. SPE-117274-PA 14(3), 393–412 (2009). https://doi.org/10.21188/117274-PA
Carrassi, A., Bocquet, M., Bertino, L., Evensen, G.: Data Assimilation in the Geosciences: An overview on methods, issues and perspectives. Wires Clim. Change 9(5), 50 (2018). https://doi.org/10.1002/wcc.535
Chen, Y., Oliver, D.S.: Levenberg-Marquardt forms of the iterative ensemble smoother for efficient history matching and uncertainty quantification. Computat. Geosci. 17, 689–703 (2013)
Chen, Y., Oliver, D.S.: History matching of the Norne full-field model using an iterative ensemble smoother. SPE Reserv. Eval. Eng. 17(2), 244–256 (2014)
Emerick, A.A.: Analysis of performance of ensemble-based assimilation of production and seismic data. J. Petroleum Sci. Eng. 139, 219–239 (2016)
Emerick, A.A., Reynolds, A.C.: Ensemble smoother with multiple data assimilation. Comput. Geosci. 55, 3–15 (2013)
Evensen, G.: Sampling strategies and square root analysis schemes for the EnKF. Ocean Dyn. 54, 539–560 (2004)
Evensen, G: Data Assimilation: The Ensemble Kalman Filter, 2nd edn. Springer (2009)
Evensen, G.: Analysis of iterative ensemble smoothers for solving inverse problems. Computat. Geosci. 22(3), 885–908 (2018). https://doi.org/10.1007/s10596-018-9731-y
Evensen, G.: Accounting for model errors in iterative ensemble smoothers. Computat. Geosci 23 (4), 761–775 (2019). https://doi.org/10.1007/s10596-019-9819-z
Evensen, G., Eikrem, K.S.: Strategies for conditioning reservoir models on rate data using ensemble smoothers. Computat. Geosci. 22(5), 1251–1270 (2018). https://doi.org/10.1007/s10596-018-9750-8
Evensen, G., Raanes, P., Stordal, A., Hove, J.: Efficient implementation of an iterative ensemble smoother for data assimilation and reservoir history matching. Front. Appl. Math. Stat. 5, 47 (2019). https://doi.org/10.3389/fams.2019.00047, https://www.frontiersin.org/article/10.3389/fams.2019.00047
Hanea, R., Evensen, G., Hustoft, L., Ek, T., Chitu, A., Wilschut, F.: Reservoir management under geological uncertainty using fast model update. In: SPE Reservoir Simulation Symposium, Society of Petroleum Engineers, p. 12, https://doi.org/10.2118/173305-MS (2015)
Hanea, R.G., Casanova, P., Hustoft, L., Bratvold, R.B., Nair, R., Hewson, C., Leeuwenburgh, O., Fonseca, R.M.: Drill and learn: A decision making workflow to quantify value of learning. In: SPE Reservoir Simulation Conference. Society of Petroleum Engineers (2017)
Houtekamer, P.L., Zhang, F.: Review of the ensemble Kalman filter for atmospheric data assimilation. Mon. Weather Rev. 144, 4489–4533 (2016)
Hunt, B.R., Kostelich, E.J., Szunyogh, I.: Efficient data assimilation for spatiotemporal chaos: A local ensemble transform Kalman filter. Physica D 230, 112–126 (2007)
Kitanidis, P.K.: Quasi-linear geostatistical therory for inversing. Water Resour. Res. 31(10), 2411–2419 (1995)
Leeuwenburgh, O., Chitu, A.G., Nair, R., Egberts, P.J.P., Ghazaryan, L., Feng, T., Hustoft, L.: Ensemble-based methods for well drilling sequence and time optimization under uncertainty. In: ECMOR XV-15th European Conference on the Mathematics of Oil Recovery (2016)
Lorentzen, R., Luo, X., Bhakta, T., Valestrand, R.: History matching the full norne field model using seismic and production data. SPE J. SPE-194205-MS 24, 1452–1467 (2019). https://doi.org/10.2118/194205-PA
Luo, X., Bhakta, T.: Automatic and adaptive localization for ensemble-based history matching. Petroleum Science and Engineering, p. 184. https://doi.org/10.1016/j.petrol.2019.106559 (2020)
Oliver, D., Reynolds, A., Liu, N.: Inverse Theory for Petroleum Reservoir Characterization and History Matching. Cambridge University Press, Cambridge (2008)
Oliver, D.S., He, N., Reynolds, A.C.: Conditioning permeability fields to pressure data. In: ECMOR V-5th European Conference on the Mathematics of Oil Recovery, p. 11, https://doi.org/10.3997/2214-4609.201406884 (1996)
Perrone, A., Pennadoro, F., Tiani, A., Della Rossa, E, Saetrom, J: Enhancing the geological models consistency in ensemble based history matching an integrated approach. Society of Petroleum Engineers. https://doi.org/10.2118/186049-MS (2017)
Raanes, P.N., Stordal, A.S., Evensen, G.: Revising the stochastic iterative ensemble smoother. Nonlin Processes Geophys 26, 325–338 (2019). https://doi.org/10.5194/npg-2019-10
Skjervheim, J.A., Evensen, G., Hove, J., Vabø, J.: An ensemble smoother for assisted history matching. SPE, 141929 (2011)
Wang, L., Oliver, D.S.: Efficient optimization of well drilling sequence with learned heuristics. SPE J 24(5), 2111–2134 (2019). https://doi.org/10.2118/195640-PA
Acknowledgements
This work received support from the Research Council of Norway and the companies AkerBP, Wintershall–DEA, Vår Energy, Petrobras, Equinor, Lundin, and Neptune Energy, through the Petromaks–2 DIGIRES project (280473) (http://digires.no). We acknowledge the access to Eclipse licenses granted by Schlumberger.
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Appendix: Implementation in ERT
Appendix: Implementation in ERT
The reservoir example adopts the Ensemble Reservoir Tool (ERT) available from Github https://github.com/equinor/ert.git for the management and conditioning of the model ensemble. The forward integration of the reservoir models is computed using the ECLIPSE reservoir simulator provided by Schlumberger. To be able to test the new formulation, some modifications were implemented in ERT and in the simulation job that runs ECLIPSE. The current version of ERT lacks functionality for simulating or specifying correlated measurement errors, and as most other assimilation implementations it works with a diagonal Cdd. On the other hand, the actual implementation of the update scheme uses the same methods as discussed in the previous section, and has already full functionality for handling a nondiagonal measurement error-covariance matrix, Cdd, or directly using the measurement perturbations, E, see Evensen et al. [12]. The following sections briefly discuss how ERT was modified to accommodate for correlated measurement errors and stochastic forcing of the reservoir simulations. For now, the implementation uses a case-specific implementation with communication through files, but the plan is to upgrade ERT to generally support correlated measurement errors.
1.1 A.1 Convergence of the subspace EnRML
The step-length scheme used in Algorithm 2 (i.e., the value of γ), as implemented in ERT, is the following: One defines the maximum step length t1 = 0.5, the minimum step length, t2 = 0.2, and a step-length decline from one iteration to the next t3 = 2.5, and we compute the value of γi, for iteration i, from
Here, γi follows a geometrical series starting from the value t1 in the first iteration and then reducing geometrically with the number of iterations towards t2. The formula in Eq. 49 allows us to define and test different step-length schemes easily. In real applications, it is desirable to start with a conservative step-length that works well ”in most applications.” If it turns out that the selected scheme becomes unstable, then the experiment should be restarted with a new and more conservative step-length. There is no exact theory on how much we would need to reduce the step size, but a good starting point can be to reduce t1 = 0.5 to t1 = 0.4.
The termination of the iterations can be based on the magnitude of the gradient, or one can use the relative change of the cost functions from one iteration to the next. For real reservoir applications, the time required per iteration can vary from hours to days, and the affordable number of iteration steps will always be limited. Thus, a practical procedure is to manually stop the iterations when one sees that the cost functions for the realizations are “almost” identical from one iteration to the next.
The rapid convergence of the assimilation algorithm for case IESR is illustrated in Fig. 17. Ten iterations were run, but after about five to six iterations there is only a marginal change in the parameters, and for practical applications five or six iterations should suffice for models of similar nonlinearity to the one used here.
1.2 A.2 Simulation of measurement perturbations
First, there is need of an application for simulating an ensemble of measurement perturbations for the time series of oil-production-rate (OPR), gas-production-rate (GPR), and water-production-rate (WPR), for each of the production wells. In the examples shown below, possible errors in the injection rates are ignored. The measurement perturbations can be white in time, red in time, or just a constant bias. The simulated ensemble of rate perturbations is then stored in a file. The current implementation computes perturbations for all the rates given in the ECLIPSE schedule file, no matter if they are used or not in the conditioning. The code used for simulating the measurement perturbations is available from the Github repository https://github.com/geirev/EnKF_sampling/tree/ERTOBS/ERTsamp.
1.3 A.3 Importing E in the EnRML algorithm
In the current implementation, line 5 of Algorithm 2 illustrates how the subspace EnRML subroutine in ERT reads all the simulated rate perturbations from a file and store them in E0. The new measurement perturbations replace the original measurement perturbations supplied by ERT in D (line 6) and are input to the inversion in the iteration of Wi (line 14). The product IeE0 in lines 6 and 7, extracts the new simulated perturbations corresponding to the measurements contained in D, which can typically be a subset of all the rates represented in E0. Thus, with this simple modification, it is possible to compute the EnRML update while allowing for correlated measurement errors.
1.4 A.4 Updating measurement perturbations E i
As soon as one has computed the transition matrix, Ti, for iteration i, in line 15 of Algorithm 2, it is possible to augment the measurement perturbations to the model state vector and update them according to
The beautiful property of the subspace EnRML algorithm is its independence of the model state vector, which only enters as input to the forward model. Thus, augmenting the measurement errors to the state vector does not impact the computational procedure used to obtain the transition matrix. However, when the measurement errors are input to the forward model, they will change the model predicted measurements, and thus lead to a different transition matrix.
1.5 A.5 Updating historical rates in the schedule file
In iteration i, the forward model is forced by the updated perturbations, g(Xi, Ei). Practically, one needs to update the Eclipse schedule file with new historical OPR, GPR, and WPR rates, by adding the updated perturbations to the corresponding “WCONHIST” rates. ERT handles this schedule modification by running a simple program, which is available from https://github.com/geirev/Schedule_parser.git. This program reads the updated perturbations in Ei from file and adds them to the historical rates defined in the original schedule file. ERT runs this program immediately before starting the Eclipse simulation for each realization.
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Evensen, G. Formulating the history matching problem with consistent error statistics. Comput Geosci 25, 945–970 (2021). https://doi.org/10.1007/s10596-021-10032-7
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DOI: https://doi.org/10.1007/s10596-021-10032-7