Maria Gheorghe TU Delft Bergen, May 2010

1. Non Parametric Bayesian Belief Nets (NPBBN) vs. Ensemble Kalman Filter (EnKF) in Reservoir Simulation Maria Gheorghe TU Delft Bergen, May 2010

2. Outline • Introduction • Problem description • Ensemble Kalman Filter (EnKF) • Localization • Bayesian Belief Nets (BBNs) • Continuous Non Parametric BBNs (NPBBNs) • Case study • Conclusions

3. Introduction • Reservoir simulation: uses mathematical-physical model to simulate reservoir behavior • Goal • Estimate permeability field for a reservoir simulation problem using the following comparative methods: • Ensemble Kalman Filter (EnKF) with localization • Non-parametric Bayesian Belief Nets (NPBBN)

4. Problem description We have one injector in the center and 4 producers at each corner of the reservoir simulated Simsim, TU Delft, Prof. Jansen • Ensemble Kalman Filter (EnKF) • Physical model: Two Phase Flow of Oil and Water • Measurements • Well data • Bottom hole pressure at injector • Total flow rates at producers We are interested in the permeability field K

5. Ensemble Kalman Filter (EnKF) • Solve the filtering problem: determine the probability density of the state • conditioned on the history of measurements • State vector dimension of the state vector For real applications, the dimension of state vector is much larger (e.g.) and the number of ensemble members is much smaller than the dimension of the state vector Presence of long distance spurious correlations Need to apply localization

6. Localization • Removes long distance spurious correlations • Two most common localization methods: • Covariance Localization (CL) • Local Analysis (LA) • Distance based methods : we must know the distance between a state space element and the observation point; • We apply LA to the EnKF such that only observations within a certain distance form the grid point being analyzed will impact the analysis in that grid point;

7. Bayesian Belief Nets (BBNs) • BBNs – directed acyclic graphs (DAG) • Nodes – univariate random variables (continuous or discrete) • Arcs – direct influences form parents to children Parents Children

8. The Qualitative & Quantitative parts of a BBN • The graph itself and the influences entailed by it form the qualitative part of a BBN. • Each variable might be associated with a conditional probability function of that variable given its parents in the graph, the quantitative part of the model consists of the conditional probability functions associated with the variables

9. Dynamic or Temporal BBNs • Special case of static BBNs aimed at time series modeling • Instances of a static BBN connected in discrete slices of time • The arcs between slices are from left to right, reflecting the casual flow of time • The term dynamic means that we are modeling a dynamic system not that the network is changes over time Functional relations

10. KF methods and Dynamic BBNs • Kalman Filter gives a recursive procedure for estimating the state space of a system governed by the following equations:

11. Continuous Non Parametric BBNs (NPBBNs) • Nodes: represented by random variables with continuous and invertible marginal distribution • Arcs: associated with constant (conditional) rank correlations, realized by a copula that realizes all correlations and represents independence at zero correlations • The (conditional) rank correlations together with the marginals and the copula uniquely determine the joint distribution Parents Children

12. Learning the structure of a NPBBN from Data • To learn the structure of a NPBBN that captures most of the dependencies present in database use data mining capabilities of UniNet (software developed at TU Delft) • Measure of dependence : the determinant of the correlation matrix

13. Ordinal data mining

14. Case study • We compare results of estimating the permeability field from: • EnKF (with localization) with • NPBBNs • Perform simulations for 100 ensembles • Assimilate every 60 days for a period of 420 days • Perform a twin experiment, i.e. we know the true values of the permeability field

15. 1. EnKF with localization

16. 2. NPBBNs • We have one injector in the middle from which we inject the water and 4 producers located in the 4 corners of the reservoir • We are interested in the joint distribution of the parameters from 9 locations

17. Time 1 • We run the model for 60 days using Simsim. The Joint distribution resulted is used to construct a saturated NPBBN using UniNet

18. Time 1 • We run the model for 60 days using Simsim. The Joint distribution resulted is used to construct a saturated NPBBN using UniNet

19. Time 1- Correlation matrix

20. Time 1- assimilation • Instead of using EnKF to assimilate the data, we conditionalize the joint distribution on observed bottom hole pressure at the injector well and total flow rates at the producers

21. Time 7 • A new NPBBN is obtained after 420 days. It is further conditionalized on the available measurements at the wells

22. Time 7 • A new NPBBN is obtained after 420 days. It is further conditionalized on the available measurements at the wells

23. Correlations • The correlations between permeability and the other variables, time1 time 3time 7

24. Ordinal data mining- time 7 • Assumption: Joint Normal Copula satisfied

25. Ordinal data mining – time 7 • We remove arcs corresponding to the small values of the (cond.) rank correlations

26. Ordinal data mining – time 7 • Validate that the BBN with its conditional independence relations is an adequate model of the saturated graph

27. Comparative results

28. Preliminary conclusions • We presented EnKF (with localization) and NPBBNs that give comparative results for estimating the permeability field in a reservoir simulation problem; • EnKF: • Requires (assumes) all conditional probability distributions to be linear Gaussian • Localization methods for EnKF can improve the estimate • NPBBNs • Allow more flexibility for the conditional probability distributions; • Is possible to apply ordinal data mining---showed to improve the estimate

29. Questions ?????

30. Thank you! 