Norges teknisk-naturvitenskapelige universitet Faculty of Engineering Science & Technology

DEPARTMENT OF PETROLEUM ENGINEERING AND APPLIED GEOPHYSICS

Reservoir Parameter Estimation for Reservoir Simulation using Ensemble Kalman Filter (EnKF) Nasima Begum M. Sc. Student Supervisor: Professor Jon Kleppe, IPT

IPT – September 2009

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Abstract For matching current production data, Ensemble Kalman Filter (EnKF) is very efficient for real-time updating of a reservoir model. By using the Ensemble Kalman Filter, assimilation of the ensemble of reservoir models for the most current observation data can be possible. Thus the estimations of reservoir model parameters, their associated uncertainty and the forecast are made up to date by this model. In this work, Ensemble Kalman Filter (EnKF) is used which has been developed from Kalman Filter to handle large, nonlinear models. The EnKF algorithm has been applied to one synthetic model and three different cases of semi-synthetic model from the Norne oil field, offshore Norway. In the EnKF method, static parameters such as porosity and permeability and dynamic variables like fluid saturation and pressure are continuously updated to match with the real- time production data for history matching purposes. It is shown that the formulation used in the Ensemble Kalman Filter reduces a nonlinear minimization of the objective function associated with an optimization problem for different models. For reservoir management, it is an important issue to utilize the available data in order to make improved forecasts. An updated model is used for forecasting where initial values are consistent with the measurements. The Kalman filter technique is used for this approach that incorporates the information from the measurements into the current state of the model, considering uncertainty that belongs both to the state of the model and the measurements where the uncertainty of the model updated simultaneously with the model itself. Compare with the usual history matching, this approach gives better agreement with the initial values for the forecast. In this work, Ensemble Kalman Filter (EnKF) has been used for estimation of the reservoir parameters like porosity and permeability from production data.

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Disclaimer The views expressed in this thesis work are he views of the author and do not necessarily reflect the views StatoilHydro and the Norne field license partners.

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Acknowledgments The author of this thesis is grateful to Professor Jon Kleppe, Department of Petroleum Engineering and Applied Geophysics, NTNU, Professor Bjarne Anton Foss, Department of Engineering Cybernetics, NTNU and Mohsen Dadashpour, PhD student, Department of Petroleum Engineering and Applied Geophysics, NTNU for their invaluable academic guidance and continuous encouragement, without which the report would not have been completed. The author extends her heartiest thanks to them for their devoted time during the preparation of this thesis, even in their busy schedule. Special thanks are extended to Jan Ivar Jensen, Erlend Våtevik and Lars Johan Sandvik for their kind support, valuable discussion and help with the IT support. The author would like to extend her heartiest gratitude to StatoilHydro (operator of the Norne field) and its licensed partners Petoro AS and Eni Norge AS and the NTNU/CIO for providing the data in all reports, publications and presentations resulting from the project. Also special thanks have been given to the Center for Integrated Operations at NTNU and StatoilHydro for the release of the Norne field data. Heartfelt thanks are extended to the International Research Institute of Stavanger (IRIS) for helping and providing with the code of Ensemble Kalman Filter (EnKF). The author is also grateful to Schlumberger-GeoQuest for their help to use of Eclipse simulator. The author extends her special thanks to the concerned authority, the Stanford University, USA for the Stanford Geostatistical Modeling Software for running Sequential Gaussian Simulation for the present study. Last but not the least, I sincerely thank my family at home (Bangladesh) and the Director General, Geological Survey of Bangladesh, who gave me all support for carrying out my higher study abroad. Trondheim, September 2009 Nasima Begum

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Contents 1 Introduction 1 1.1. Background ……………………………………………………………… 1 1.2. Previous work …………………………………………………………… 2 1.3. Objectives ……………………………………………………………….. 4 2 History Matching 5 2.1. Introduction ............................................................................................... 5 2.2. Manual versus Automated history matching ……………………………. 6 2.3. Limitations of history matching …………………………………………. 6 2.4. New challenges of history matching …………………………………….. 6 2.5. Automatic history matching ....................................................................... 7 2.5.1. Simulated annealing ....................................................................... 8 2.5.2. Genetic algorithm ........................................................................... 9 2.5.3. Polytope ......................................................................................... 9 2.5.4. Scatter and Tabu searches .............................................................. 10 2.5.5. Neighborhood algorithm ................................................................ 10 2.5.6. Kalman filter ……………………………………………………… 11 2.6. Differences between traditional history matching and EnKF …………… 12 2.7. Parameter estimation …………………………………………………….. 12 2.7.1. Construct a mathematical method ……………………………….. 13 2.7.2. Define an objective function …………………………………….. 13 2.7.3. Optimization method ……………………………………………. 15 2.8. Quantifying uncertainty in production forecasts ………………………… 17 2.8.1. Errors induced …………………………………………………… 17 2.8.2. Sensitivities ……………………………………………………… 18 2.8.3. Selection of meaningful parameters …………………………….. 18 2.9. Methods for uncertainty analysis ……………………………………….. 19 2.9.1. Linearization method …………………………………………… 19 2.9.2. Bayesian approach ……………………………………………… 20 3 Statistic Fundamentals 22 3.1. Random variables ……………………………………………………….. 22 3.2. Probability density function …………………………………………….. 23 3.3. Expected value, Variance and Covariance ……………………………… 23 3.3.1. Expected value ………………………………………………….. 23 3.3.2. Variance ………………………………………………………… 24 3.3.3. Covariance ……………………………………………………… 24 3.4. Probability distributions ………………………………………………… 24 3.5. Approximations using samples from a distribution …………………….. 25

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4 Theoretical Formulation of the EnKF 27 4.1. Linear Kalman filter ………………………………………………...... 27 4.1.1. Problem formulation ………………………………………….. 27 4.1.2. Kalman filter equations ………………………………………… 28 4.2. Ensemble Kalman filter ………………………………………………. 30 4.2.1. Problem formulation ………………………………………….. 30 4.2.2. Ensemble covariance …………………………………………. 31 4.2.3. Ensemble representation ……………………………………… 31 4.2.4. EnKF algorithm ………………………………………………. 32 4.3. Theoretical comparison of linear Kalman filter (KF) and EnKF ……. 35 4.4. EnKF methodology …………………………………………………... 36 4.5. Managing the uncertainties …………………………………………… 36 4.6. Advantages and disadvantages of EnKF …………………………….. 37 4.7. Use of EnKF for reservoir parameter estimation ……………………. 38 4.7.1. Steps used for EnKF …………………………………………. 38 4.7.2. Steps used for parameter estimation …………………………. 39 4.7.3. Measures of filter performance ………………………………. 41 5 SGSIM 42 5.1. Fundamentals ………………………………………………………… 42 5.2. Sequential simulation ………………………………………………… 43 5.3. Estimating the local conditional distributions ……………………….. 44 5.4. Kriging ………………………………………………………………. 47 5.4.1. Indicator kriging (IK)………………………………………… 48 5.4.2. Simple kriging (SK)………………………………………….. 49 5.4.3. Ordinary kriging (OK)……………………………………….. 49 5.4.4. Non-Linear kriging …………………………………………... 49 5.5. Sequential Gaussian simulation ……………………………………… 50 6 Results and Discussion 53 6.1. Case study ……………………………………………………………. 53 6.1.1. SPE Comparative 3 …………………………………………… 54 6.1.2. Norne Field …………………………………………………… 54 6.1.2.1. General field information …………………………... 54 6.1.2.2. General geology of Norne field …………………….. 56 6.1.2.3. Structure of the Norne field ………………………… 58 6.1.2.4. Depositional environment ………………………….. 59 6.1.2.5. Production history of the Norne field ……………… 59 6.1.2.6. Oil enhancement techniques ………………………. 59 6.2. Synthetic and semi-synthetic case …………………………………… 60 6.2.1. Case 1 ………………………………………………………... 60 6.2.2. Norne ………………………………………………………… 64 6.3. Ensemble Kalman filter for this study ………………………………. 69 6.3.1. Introduction ………………………………………………….. 69 6.3.2. Input files ……………………………………………………. 70 6.3.3. Output files ………………………………………………….. 72

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6.4. Discussion of the cases on the basis of EnKF ……………………… 76 6.4.1. Case 1 ……………………………………………………….. 76 6.4.2. Norne ………………………………………………………... 81 6.4.2.1. Case A …………………………………………….. 81 6.4.2.2. Case B …………………………………………….. 82 6.4.2.3. Case C …………………………………………….. 82 7 Summary and Recommendations 109 References 113 Symbols 123 Nomenclature 126 Appendix A 127 Appendix B 133

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List of Tables 1 Gas-oil and oil-water contacts in the Norne Field ……………………………. 58 2 Recoverable and remaining reserves in the Norne Field [NPD, (2009)] …….. 59 3 Reservoir Grid and Saturation Input Data for Case 1 ………………………… 60 4 Well Input Data for Case 1 …………………………………………………… 61 5 Reservoir Grid and Fault Data ……………………………………………….. 65 6 Well Input Data for Norne …………………………………………………… 66 7 Standard Deviation and Error Estimation for Case 1 ………………………… 107 8 Standard Deviation and Error Estimation for Norne Three Cases …………… 107

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List of Figures 6.1 a) Location of Norne field in respect with Heidrun field in Norwegian Sea

& b) Location of the Norne oil field in the Norwegian Sea ………………… 55 6.2 The Norne Field segments and wells ……………………………………….. 56 6.3 Stratigraphical sub-division of the Norne reservoir ………………………… 57 6.4 NE-SW running structural cross section through the Norne field with initial and indications of present fluid contacts, and current drainage strategy …… 58 6.5 a) Reservoir model grid and b) Well perforations for Case 1 ……………… 62 6.6 a) Reference or real porosity and b) Real permeability for Case 1 ………… 63 6.7 Initial guess of reservoir pore pressure a) Beginning and b) End of production .. 64 6.8 Initial guess of reservoir water saturation a) Beginning and b) End of production 64 6.9 Cross-section of the Norne Field in 50th cell ……………………………….. 65 6.10 a) Reservoir grid and major faults and b) well perforations of Norne Field … 67 6.11 a) Real porosity and b) Real permeability for Norne reservoir model ……… 68 6.12 Initial guess of reservoir pore pressure a) Beginning and b) End of production 69 6.13 Initial guess of reservoir water saturation a) Beginning and b) End of production 69 6.14 IRIS - EnKF Matlab work flow description ………………………………… 73 6.15 Preparing the Matlab input files. Work flow of setupCase.m ……………… 74 6.16 Description of the main script ……………………………………………… 75 6.17 Compute reference solution ………………………………………………… 76 6.18 True static and reference field of porosity and permeability for Case 1……. 77 6.19 Estimated static fields for porosity and standard deviation for Case 1 …….. 78 6.20 Estimated static fields for permeability and standard deviation for Case 1 ... 79 6.21 a) True well measurements used in the Ensemble Kalman updating,

b) Reference solution and forecasts with initial members for the wells and c) Reference solution and forecasts with estimated ensemble members for Case 1 80

6.22 Some of multiple realizations of (a) porosity and (b) permeability ……….. 82 6.23 Comparison of the true static and reference field of porosity and

permeability for Norne …………………………………………………….. 83 6.24 Comparison of the estimated static fields for porosity and standard

deviation for Norne ………………………………………………………... 84 6.25 Comparison of the estimated static fields for permeability and

associated standard deviation (SD) for Norne …………………………….. 85 6.26 True well measurements used in the Ensemble Kalman updating for Norne 86 6.27 Reference solution and forecasts with initial members for three cases of Norne 88 6.28 Reference solution and forecasts with estimated ensemble members for Norne 89 6.29 Comparison of the real and mean porosity for Case 1 …………………… 90 6.30 Comparison of the real and mean permeability for Case 1 ……………… 90 6.31 Comparison of the real and mean porosity for Norne …………………… 91 6.32 Comparison of the real and mean permeability for Norne ………………. 91 6.33 Cumulative oil production where acceptable simulations (broken lines)

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were used to forecast future productions of next 10 years for Case 1 …. 92 6.34 Cumulative bottom hole pressure with forecast for Case 1 …………… 93 6.35 Cumulative well water cut with forecast for Case 1 ………………….. 93 6.36 Cumulative oil production for Norne Case A where acceptable simulations

(broken lines) were used to forecast future productions of next 10 years … 94 6.37 Cumulative oil production with forecast for Norne Case B …………… 95 6.38 Cumulative oil production with forecast for Norne Case C …………… 95 6.39 Comparative figures show cumulative bottom hole pressure with forecast

for injection and production well in all the three cases of Norne ……… 96 6.40 Cumulative mean oil production with forecast compare to true

and initial data for Case 1 ………………………………………………. 97 6.41 Cumulative mean water production with forecast compare to

true and initial data for Case 1 ………………………………………….. 98 6.42 Cumulative mean gas production with forecast compares to true

and initial data for Case 1 ……………………………………………….. 98 6.43 Mean cumulative bottom hole pressure for Case 1 …………………….. 99 6.44 Mean cumulative water cut for Case 1 …………………………………. 99 6.45 Cumulative mean oil production with forecast for Case A,

B and C for Norne comparing with the true and initial values …………. 100 6.46 Cumulative mean water production with forecast for Norne …………... 101 6.47 Mean cumulative bottom hole pressure with forecast of the

injection well for Norne ………………………………………………… 101 6.48 Mean cumulative bottom hole pressure with forecast of the

production well for Norne ……………………………………………… 102 6.49 Comparison of calculated lowest objective function and mean

values together with forecast in oil production, well bottom hole pressure and water cut for Case 1 ……………………………………… 103

6.50 Comparison of lowest objective function and mean values with forecast in oil production, water production and well bottom hole pressure of injection and production wells for Norne ……. 104 - 105

6.51 Objective function with the number of time steps for Case 1 …………. 106 6.52 Objective function with number of time steps showing comparison

of three different cases from Norne (Case A, B & C) …………………. 107

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Chapter 1 Introduction 1.1. Background Simulation is the imitation of real things, state of affairs, or a process. The act of simulating something generally entails representing certain key characteristics or behavior of a selected physical or abstract system. It is used in many contexts, including the modeling of natural systems or technology for performance optimization. It can be used to show the eventual real effects of alternative conditions and courses of action. Key issues in simulation include acquisition of valid source information about the referent, selection of key characteristics and behavior, the use of simplifying approximations and assumptions within the simulation, and fidelity and validity of the simulation outcomes. Simulation reproduces histogram, honors spatial variability (variogram). It is appropriate for flow simulation and allows an assessment of certainty with alternate possible realizations. Reservoir simulation is an area of reservoir engineering in which computer models are used to predict the flow of fluids (typically oil, water and gas) through porous media. Reservoir simulation models are used by oil and gas companies in the development of new fields as well as in the developed fields where production forecasts are needed to help make investment decisions. As building and maintaining a robust, reliable model of a field is often time-consuming and expensive; models are typically only constructed where large investment decisions are at stake. Improvements in simulation software have reduced the time to develop a model. Models give good predictive abilities which are the important asset in reservoir management. Models are based on various data like production data, seismic data with full of uncertainties. The predictive value of the models needs updating as they tend to deteriorate over time. Thus, model updating at regular intervals are required which is known as History Matching.

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History Matching is a well-established field in reservoir engineering. It is not only mathematically or computationally challenging, but also non-unique. It is a time consuming work which was done manually in the beginning, but now a lot of research is done to automate this procedure. Ensemble Kalman Filter (EnKF) is a new method of automatic history matching. This method was first introduced by Evensen in 1964 to handle the large nonlinear oceanic models. The first history matching application was given by Nævdal, et al. in 2002. Historically, the Kalman filter (Kalman, 1960) is the most widely used sequential data assimilation method applicable to linear system. As the reservoir simulation equations are highly nonlinear when multiple phases co-exist, the Kalman filter is inappropriate for the typical automatic history matching problems in reservoir characterization. In 1994, Evensen introduced the Ensemble Kalman Filter (EnKF) which can be applied to the nonlinear systems. The EnKF is independent of reservoir simulators and does not require the adjoint code. It outputs a set of estimated models, which are suitable for uncertainty analysis. To improve the predicted capability of the models and estimate the poorly known parameters, the Ensemble Kalman Filter has recently been used with simulation models for oil and gas reservoirs. On the basis of the information contained in pressure and the rate measurement from the production wells, the reservoir model state and parameters are updated sequentially with time by using EnKF and is suitable for real-time reservoir history matching. The Ensemble Kalman Filter is a Monte Carlo approach where the parameters are updated sequentially according to the chronological order in which the data are acquired and assimilated. It is a non-gradient based history matching method that does not require the tedious derivation and implementation like gradient-based techniques. The Ensemble Kalman Filter has three steps: initial sampling, forecasting through a reservoir simulator and assimilation. The initial random sampling with less dependence ensemble members achieves the same level of accuracy through a small ensemble size. In automatic history matching, the independence of the ensemble members gives high-performance computation during the forecasting step. In order to achieve a significant speed-up, two-level computation is adopted; distributing ensemble members simultaneously while simulating each member in a parallel style. 1.2. Previous work Several articles about the successful application of the Ensemble Kalman Filter have been published in the petroleum industry. Research on Ensemble Kalman Filtering (EnKF) started with Evensen (1994) and later by Houtekamer and Mitchell (1998) where this method is classified as perturbed observations of EnKF and are essentially ensembles of data assimilation systems. Another type of EnKF is a class of square root filters (Anderson, 2001, Whitaker and Hamill, 2002, Bishop, et al. 2001, and Tippett, et al. 2003). This consists of a single analysis based on the ensemble mean, and the analysis perturbations are obtained from the square root of the Kalman Filter analysis error covariance. Whitaker and Hamill, (2002) showed that square root filters are more accurate than perturbed observation filters because of sampling errors introduced by perturbing the

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observations with random errors. The square root filters discussed by Tippett, et al. (2003) assimilate observations sequentially (Houtekamer and Mitchell, 1998) which increases efficiency by avoiding the inversion of large matrices. Nævdal, et al. (2002) used EnKF for permeability estimation on a simple reservoir. In their work EnKF was used to update static parameters in near-well reservoir models by tuning the permeability field. Nævdal, et al. (2005) used EnKF to estimate the permeability of the whole reservoir using a simplified 2D reservoir model of a North Sea field. In this work, they updated the two-dimensional three-phase reservoir model by continuously adjusting the static permeability field and dynamic saturation and pressure fields at each assimilation steps. Gu and Oliver (2005) examined the use of EnKF for state and parameter estimation on the PUNQ-S3 reservoir test case. Here EnKF is used to update the porosity and permeability fields, as well as the saturation and pressure fields, and then applied it to match the three-phase production data for the three-dimensional PUNQ-S3 reservoir model. Brouwer, et al. (2004) investigated EnKF in a closed loop setting with optimal reservoir control in a simple two-dimensional model. Nævdal, et al. (2006) used the combination of EnKF for continuous model updating with an automated adjoint-based water flood optimization to optimize water flooding strategy. Gao, Zafari and Reynolds (2005) compared EnKF with another method called Randomized Maximum Likelihood (RML). One major difference between the methods is that RML is an adjoint based method whereas the EnKF does not require much effort in coupling with the reservoir model. Zafari and Reynolds (2005) tested the method on some nonlinear problems to validate the EnKF. EnKF was proven to have difficulties in multi-modal distributions and that the Gaussian assumption of EnKF is very critical. Wen and Chen (2005) presented a modified version of EnKF where they added a “confirming” step to run reservoir simulation using most recent updated static model parameters so that the updated static and dynamic parameters are always consistent. However Zafari and Reynolds (2005) used a linear case to show that the improved algorithm of “confirming” step suggested by Wen and Chen (2005) is inconsistent. Liu and Oliver (2005a and 2005b) used the EnKF both for history matching and for facies estimation in a reservoir simulation model. They found that the EnKF method outperformed the gradient-based minimization method in both computation efficiency and applicability. Lorentzen, et al. (2005) studied the robustness of EnKF by running ten ensemble cases of different initial conditions where they found EnKF is well-suited for forecasting uncertainty. Skjervheim, et al. (2005) suggested a method based on EnKF to incorporate 4D seismic data in continuous model updating. They showed that the EnKF could handle large amount of seismic data and had a positive impact on matching the permeability field from very noisy measurement data. Dong, et al. (2006) also identified a similar conclusion to Skjervheim, et al. (2005) by using the EnKF for reservoir description to history match in both production and time-lapse 4D seismic data. Reinlie (2006) used the traditional EnKF and further conditioned local permeability information around the well bore. Lorentzen, et al. (2006) used EnKF as an optimization routine for controlling down-hole chokes in smart wells with the aim of optimizing water flooding. Haugen, et al. (2006) tested EnKF for history matching on a North Sea field case by using real production data. They showed how EnKF can improve the model parameters for history matching and also discussed the updating of the reservoir states. They showed promising results, but concluded that the

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work should be done to be able to estimate other reservoir parameters using the EnKF. Park and Choe (2006) studied the low value of the estimate error covariance after some history matching periods and the number versus quality of the measurement data. They suggested a regeneration step when the estimate error covariance reaches one fifth level of the initial estimate error covariance. They found the water saturation measurements near the irreducible water saturation or the residual oil saturation are not sensitive to reservoir static parameters and can be ignored. This ignorance and measurement selection helps to avoid ensemble deviation and improves the history matching of reservoir porosity and permeability. Streamline covariance localization and streamline assisted tool in EnKF was proposed by Arroyo-Negrete (2006). Evensen, et al. (2007) also showed how EnKF can be used in assisting history matching of a North Sea reservoir model. They investigated more parameters such as WOC and GOC, vertical transmissivity multipliers and fault transmissivity multipliers. They also discussed non-Gaussian parameter distribution and pointed out that the EnKF is theoretically unrealistic if it is used directly on a multimodal prior, such as a reservoir consisting of channels. 1.3. Objectives The objective of this study is the estimation of porosity and permeability from production data using ensemble Kalman filter (EnKF). Tasks:

Ensemble Kalman Filter methodology will be comprehensively studied by integrating production measurements and geological model information for continuous reservoir model updating

Investigate various assimilation schemes for the assimilation step in the Ensemble Kalman Filter methodology

Use Ensemble Kalman Filter for history matching and quantify the forecasting uncertainty for different cases (synthetic and semi-synthetic)

Find out the average volumetric error in estimation of porosity and permeability in different conditions

Implement and test method using simple ECLIPSE 100 simulation model to represent the true reservoir and simulate this with a production and injection plan

Then implement an EnKF and investigate how well the method works

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Chapter 2 History Matching 2.1. Introduction The process of adjusting the variables in a reservoir simulation model to match observations of rates, pressures, saturations, and other variables for individual wells is called history matching. In many cases, general geological information also needs to be honored, such as the variance-covariance structure of the model parameters (Yaqing & Oliver, 2005). History matching requires the minimization of the square of the mismatch between all measurements and computed values and also of the current and prior model parameters. It also requires numerous iterations or runs which make the procedure very costly and time consuming. Historical production data contain reservoir information that will reduce the uncertainty in the future performance. Assessment of uncertainty is important in a good history match. The method of estimating the uncertainty would be to generate multiple initial reservoir models, history match each one of them and simulate for future production on all of them. In this study work, Ensemble Kalman Filter is used which is an automatic history matching method to estimate the uncertainty in the future performance. By using the Ensemble Kalman Filter, the model can be used to forecast production for the next few years with some uncertainties. The up-gradation of the model is done by collecting updated new data with reduced uncertainty. In this way, the process of history matching does not end before the life of the field ends. In history matching a probabilistic approach is used and the Bayesian approach is one of them. Here the model parameters are regarded as stochastic and the reservoir model is described with a probability density function (pdf).

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2.2. Manual versus Automated history matching During manual history matching, manually changing one or two parameters by trial-and-error can be tedious and inconsistent with the geological models. To make the parameters best fit with the simulated and observed data gives considerable uncertainties and does not have the reliability for a longer period. Automated history matching is much faster and requires fewer simulation runs than manual history matching. It includes a large number of different parameters and tackles a large number of wells without problems while in manual history matching one or two parameters are varied at time and it would require preliminary analysis first for tackling the wells. Also automatic history matching could give more reliable results in the case of complex lithology conditions with considerable heterogeneity. The basic process in automatic history matching is to start from an initial parameter guess and to improve it by integrating field data in an automatic loop. Here parameter changes are done by computer programming to minimize the function to show differences between simulated and observed data. This is called objective function that includes both model mismatch and data mismatch parts. 2.3. Limitations of history matching History matching is not only difficult but it is also a non-unique problem that means we encourage with a very big solution space. The best fit for the production data in some cases does not give good prediction for the future production of the field as production forecast is after all the main goal of history matching. Sometimes a model with parameters close to those of the base case might lead to a good forecast for the true case though the production match is bad (Tavassoli, et al. 2004). The quality of input data is essential for history matching. Measurements of gas and water do not have the desired precision when oil is the primary fluid of interest. 2.4. New challenges of history matching Today non-gradient based algorithms such as simulated annealing and the genetic algorithms are used for history matching simulation. With the help of these algorithms history matching can be possible without any access to the simulation source code or having specific knowledge of the simulator. One intriguing possibility of these algorithms is the Ensemble Kalman Filter (EnKF).

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2.5. Automatic history matching Various investigations were made about many methods of automatic history matching for the last few decades in the petroleum sector. These methods can be classified as:

a) Deterministic Algorithm b) Stochastic Algorithm

a) Deterministic algorithm Deterministic algorithms use traditional optimization approaches and obtain one-local optimum reservoir model within the number of simulation iteration constraints. In implementation, the gradient of objective function is calculated and the direction of the optimization search is then determined (Liang, 2007). The gradient based algorithms minimize the difference between the observed and simulated measurements which is called the minimization of the objective function that considered the following loop:

To run the flow simulator for the complete history matching period, To evaluate the cost function, To update the static parameters and go back to the first step.

The following is the list of several algorithms that are commonly used for the basis of gradient based algorithms (Landa, 1979 & Liang, 2007):

Gradient based algorithms: • Steepest Descent • Gauss-Newton (GN) • Levenberg-Marquardt • Singular Value Decomposition • Conjugate Gradient • Quasi-Newton • Limited Memory Broyden Fletcher Goldfarb Shanno (LBFGS) • Gradual Deformation

b) Stochastic algorithm The stochastic algorithm takes considerable amounts of computational time compared to a deterministic algorithm, but due to the rapid development of computer memory and computation speed, stochastic algorithms are receiving more and more attention. Stochastic algorithms have three main direct advantages:

The stochastic approach generates a number of equal probable reservoir models and therefore is more suitable to non-unique history matching problems,

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It is straight-forward to quantify the uncertainty of performance forecasting by using these equal probable model,

Stochastic algorithms theoretically reach the global optimum.

Following is the list of several algorithms that are commonly used on the basis of non-gradient based stochastic algorithms (Landa, 1979 & Liang, 2007):

Non-gradient based algorithms: • Simulated Annealing • Genetic Algorithm • Polytope • Scatter & Tabu Searches • Neighborhood • Kalman Filter

2.5.1. Simulated annealing Simulated annealing is a probabilistic algorithm for global optimization problems, specifically locating a good approximation to the global optimum of a given function in a large search space. For certain problems simulated annealing finds an acceptably good solution in a fixed amount of time. The name comes from annealing in metallurgy, a technique involving heating and controlled cooling of material to increase the size of its crystals and reduce their defects. The system at any time is approximately in thermodynamic equilibrium. The heat causes the atoms to become unstuck from their initial positions and wander randomly through states of higher energy. The slow cooling gives those more chances of finding configurations with lower internal energy than the initial one. As cooling proceeds, the system becomes more ordered and approaches a "frozen" ground state at T=0. By analogy with this physical process, each step of simulated annealing replaces the current solution by a neighbor, chosen with a probability that depends on the difference between the corresponding function values (Liang, 2007). Simulated Annealing is a generalization of a Monte Carlo method for examining the equations of state and frozen states of n-body systems (Metropolis, et al. 1953). The original Metropolis scheme was that an initial state of a thermodynamic system was chosen at energy E and temperature T, holding T constant the initial configuration is perturbed and the change in energy dE is computed. If the change in energy is negative, the new configuration is accepted. If the change in energy is positive, it is accepted with a probability given by the Boltzmann factor exp -(dE/T). This processes is then repeated sufficient times to give good sampling statistics for the current temperature, and then the temperature is decremented and the entire process repeated until a frozen state is achieved at T=0. By analogy the generalization of this Monte Carlo approach to combinatorial problems is straight-forward (Kirkpatrick, et al. 1983 and Cerny, 1985). The current state of the thermodynamic system is analogous to the current solution of the combinatorial problem, the energy equation for the thermodynamic system is analogous to the objective function, and ground state is analogous to the global minimum. The major difficulty (art)

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in implementation of the algorithm is that there is no obvious analogy for the temperature T with respect to a free parameter in the combinatorial problem. Furthermore, avoidance of entrainment in local minima (quenching) is dependent on the "annealing schedule", the choice of initial temperature, how many iterations are performed at each temperature, and how much the temperature is decremented at each step as cooling proceeds. Panda and Lake (1993), and Portellaand and Prais (1999) have applied the simulated annealing technique to reservoir history matching. 2.5.2. Genetic algorithm Genetic algorithms search around for the best combination of parameters for the further improvement of the match and reject the bad ones. It is a biological principle of evolution. Genetic Algorithms (GA) are adaptive heuristic search algorithm premised on the evolutionary ideas of natural selection and genetics. The method involves a population of chromosomes where each chromosome is typically encoded as a bit string and processed by “natural selection” of one generation to the next generation, associated with inheritance, mutation, selection and crossover. The possible solutions are called individuals. The basic concept is designed to simulate processes in natural system necessary for evolution, specifically those that follow the principles first laid down by Charles Darwin of survival of the fittest. As such they represent an intelligent exploitation of a random search within a defined search space to solve a problem. First pioneered by John Holland in the 60s, at the University of Michigan, genetic algorithms has been widely studied, experimented and applied in many fields in the world of engineering (Holland, 1975). Many of the real world problems involved finding optimal parameters, which might prove difficult for traditional methods but ideal for the genetic algorithm. It is still very limited in real problems due to the rising of the computational cost from the slow convergence. Williams, et al. (2004) proposed BP’s “Top-Down Reservoir Modeling” approach and was implemented by Kromah, et al. (2005). In this approach genetic algorithm is used as a global optimizer in conjunction with the reservoir simulator to achieve flexible and scaleable history matching and uncertainty quantification. The genetic algorithms solve the problem of minimizing the objective function where the problem is in a large scale with many local minima. But in the other hand, there are no guarantees in the solution of this algorithm. 2.5.3. Polytope Polytope is a generic term used to mean a number of related, but slightly different mathematical objects. Coxeter (1973) defines polytope as the general term of the sequence “point, line segment, polygon, polyhedron …” or more specifically as a finite region of -dimensional space enclosed by a finite number of hyperplanes. The word "polytope" was introduced by Alicia Boole Stott, daughter of logician George Boole (MacHale, 1985). For history matching in the petroleum industry, the polytope method is used as a multidimensional method for both efficiency and robustness (Press, et al. 1989). In the polytope method, one starts from K+1 sets of rDk values, rD1…rDK , where K is equal to the

10

number of regions. K+1 initial realization is generated and flow simulation is completed for all K+1 sets. This constitutes the initial guess for the polytope method (initial members of the polytope). Out of the K+1 set, the set with the highest error (worst match) is perturbed into a new set of rDk values. Flow simulation is completed and a new error is calculated for this new set. The previous unchanged sets of rDk plus this new set become the new members of the polytope. This is repeated until the method has converged to a minimum. Since K+1 initialization runs are needed and the K parameters must be updated dependently, this method becomes very time-consuming as K increases. The polytope method converged extremely fast in some realizations, but in others, it could not converge at all (i.e., it is not robust for this type of problem). Therefore, it will not be considered in subsequent examples. 2.5.4. Scatter & Tabu searches Scatter and Tabu Searches are a parallel to genetic algorithms, established by Fred Glover at the University of Colorado (Glover, 1977). Scatter search (Glover, 1994) operates on a set of points called reference points that constitute good solutions obtained from previous solution efforts. The approach systematically generates linear combinations of the reference points to create new points, each of which is mapped into associated feasible points. Tabu Search (Glover and Laguna, 1997) is an intelligent guidance for the search process in order to screen certain solutions from being chosen on the basis of information that suggests these solutions may duplicate of significantly resemble previous solutions. Such screening is often done by defining suitable attributes of moves or solutions, and by imposing restrictions on a set of attributes according to the search history. Recency and frequency are the two prominent techniques for exploiting search history in Tabu search. Recency is a short-term memory managed by structures or arrays called “tabu lists”, while frequency memory is more often fulfills long term search functionality. In the petroleum industry, for petroleum exploration and production, an optimizer containing scatter search, tabu search and neural networks into some simple examples were used by April, et al. (2003a) and it also used for portfolio management (April, et al. 2003b). Cullick, et al. (2003) used such an optimizer in multiple field scheduling and production strategy. Sousa, et al. (2006) used scatter search for simple history matching cases where Cullick, et al. (2006) combined scatter search with nonlinear neural network proxy for history matching problems with a small number of unknown parameters. 2.5.5. Neighborhood algorithm The neighborhood algorithm is a stochastic optimization algorithm initially aimed for seismic inversion problems (Sambridge, 1999a, 1999b) that is now used for history matching (Christie, et al. 2002 and 2006; Suzuki and Caers, 2006). It is a popular direct search inversion technique. It finds models of acceptable data in a multidimensional parameter space similar to simulated annealing and genetic algorithms. For dispersion curve inversion, physical conditions between parameters sV and pV (linked by Poisson's

11

ratio) may limit the parameter space with complex boundaries. Other conditions may come from prior information about the geological structure. For problems affected by non-uniqueness, the ideal solution is made of the ensemble of all models that equally fits the data and prior information. Wathelet (2008) shows a dynamic scaling of the parameters during the convergence to the solutions drastically improves the exploration by exploiting the properties of the Voronoi cells. 2.5.6. Kalman filter The Kalman Filter (KF) was developed by Kalman (1960) is the best known sequential data assimilation scheme. The Kalman Filter is a set of mathematical equations that provides an efficient computational (recursive) means to estimate the state of a process, in a way that minimizes the mean of the squared error. The filter is very powerful in several aspects: it supports estimations of past, present, and even future states, and it can do so even when the precise nature of the modeled system is unknown. It estimates the state of a linear dynamic system from a series of noisy measurements. In estimation theory, the Extended Kalman Filter (EKF) is the nonlinear version of the Kalman Filter which linearizes about the current mean and covariance. The disadvantage of EKF is that it is rather inefficient in case of very nonlinear systems as explained in Julier and Uhlman (1996). Moreover the method is not suited for large dimension systems, as the calculation of the derivatives, using a finite difference method, demands n + 1 model evaluations for each time step (n is the dimension of the state vector), and q + 1 evaluations of the observation operator (q is dimension of the observation space). The other possibility is to write a tangent linear model, but it is generally difficult for complex models or impossible for highly nonlinear models. The derivation of a tangent linear model to approximate a complex system may be very tedious, as well as techniques to treat the instabilities which might arise from such an approximation (Evensen, 1992). Many variations of Kalman Filter have been proposed for handle these difficulties. Among them Ensemble Kalman Filter (EnKF) is promising. The Ensemble Kalman Filter is a statistical method. It is a recursive filter that means when new data arrives, it can just compute the next step and there is no need to run a full new optimization over the horizon and is suitable for problems with a large number of variables, such as discretizations of partial differential equations in geophysical models. The EnKF originated as a version of the Kalman filter for large problems where the covariance matrix is replaced by the sample covariance and it is now an important data assimilation component of ensemble forecasting. EnKF is related to the particle filter that is same thing as am ensemble member, but the EnKF makes the assumption that all probability distributions involved are Gaussian; when it is applicable; it is much more efficient than the particle filter. The EnKF avoids computing the adjoint equations or derivatives of sensitivity coefficients by estimating the state error covariance function directly from the ensemble. Thus its implementation is very simple and independent compared to any reservoir simulator. Due to the simple formulation and easy implementation, the Ensemble Kalman Filter has

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gained popularity for weather forecasting, oceanography, hydrology and petroleum engineering. In this thesis work, non-gradient based Ensemble Kalman Filter is used for automatic history matching and parameter estimation is described in details at Chapter 4. 2.6. Differences between traditional history matching and EnKF Traditional history matching updates the static parameters like porosity and permeability and updates it until the match is reached by rerunning the model iteratively (Liang, 2007) like:

Repeated flow simulations in the entire production history, Sensitivity coefficient calculations, Not fully automated, History matching repeated with all data when new data are available, Not suitable for real-time reservoir model updating, Difficult for uncertainty assessment.

The EnKF updates the reservoir model sequentially for both static parameters like porosity and permeability and dynamic parameters like pressure and saturations:

Suitable for updating nonlinear reservoir simulation model on large scale, One flow simulation for each ensemble member, easy for distributed

computing, No need for sensitivity coefficients, Fully automated, Production data assimilated sequentially in time, Ensemble members are updated sequentially in time and reflect the latest

assimilated data, Uncertainty of prediction always up-to-date and straightforward from the

ensemble members.

Traditional history matching searches the minimum of a cost function that solves for the mode of the posterior pdf, whereas the EnKF solves the mean of the pdf as the mean is easier to estimate with small ensemble size. For this, Gaussian assumption is imposed to the pdf. Traditional history matching requires the computation of the gradient of the objective function values, whereas EnKF is not calculated the gradient that saves a lot of extra runs of the simulator. 2.7. Parameter estimation Estimation is the calculated approximation of a result which is usable if input data is incomplete or uncertain. In mathematics, approximation or estimation typically means finding upper or lower bounds of a quantity that cannot readily be computed precisely.

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Parameter Estimation is a useful application of history matching for the determination of the reservoir properties. Every method of the parameter estimation has three major steps:

1) Construct a Mathematical Method, 2) Define an Objective Function, 3) Choose a Minimization Algorithm (Optimization).

2.7.1. Construct a mathematical method A mathematical model predicts the behavior of the system under different conditions with reasonable accuracy. The problem of computing the response of the mathematical model to an external perturbation is referred to as the forward problem. The opposite problem, the inverse problem consists of finding a set of parameters for a given model such that the predicted behavior of the system replicates the true behavior (measurements) under the same set of external conditions. The following fundamental laws are relevant to the dynamics of the reservoir:

Mass conservation law, Darcy’s law, Equation of state, Relative permeability and capillary pressure relationships.

The mathematical model built by combining these laws and results in a system of differential equations (Dadashpour, 2005). 2.7.2. Define an objective function The objective function measures the discrepancy between the data and the simulator response for a given set of parameters. There are three different formulas for calculating the objective function.

a) Least squares formulation:

obs cal T obs calE ( d d ) ( d d )= − −r r r r

… … … … … … … (2.1) where, • E is the objective function • obsd

ris the historical production data

• caldr

is the calculated production data

b) Weighted least squares formulation: obs cal T obs calE ( d d ) W( d d )= − −

r r r r… … … … … … ..(2.2)

where, • W is a diagonal matrix that assigns individual weights to each measurement

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These weights for each data type and for each well assign as a function of number of data points available in a set, and on the uncertainty associated with each type of measurement.

c) Generalized least squares formulation:

*

obs priorE E Eμ= + … … … … … … … … … …(2.3) where, • obsE is the weighted sum of the squares of the differences between the simulated

production values cald and the historical production data obsd • *μ is the weighting factor which expresses the relative strength in the initial model • priorE is the prior objective function

obs cal obsn

obs 21

( d d )1E w2 σ=

−= ∑ i i

ii i

… … … … … … … .. (2.4)

where, • nobs

is the number of observations (data) • wi is the factor used to weight and equalized units

• caldi are the data calculated by the mathematical model of the reservoir

• obsdi are the measured data

• σ i is the standard deviation of measurement error of thi observation

T 1prior prior priorE ( ) C ( )αα α α α−= − − … … … ... … ..(2.5)

where, • α is the reservoir parameter • priorα is the priori mean value of the means, variances and covariance of the

parameters • Cα is the covariance matrix of the parameters of the mathematical model that gives

indications of the amount of correlations of the parameters

History matching is usually obtained by the minimization of an objective function. The objective function which is used in the minimization algorithm is:

obs cal T 1 obs cal T 1d prior prior

1 1E {( d d ) C ( d d )} {( ) C ( )}2 2 αμ α α α α− −= − − + − −

r r r r r r r r …(2.6)

where, • dC is the covariance matrix of the data that provides information about the

correlation among the data

15

• μ is the shear modulus of solid framework

Besides revealing the uncertainty associated with each parameter, the a priori covariance matrix indicates how the parameters are correlated:

d ,C 1= +ij indicates a strong correlation between the thi and thj parameter

d ,C 0=ij means no correlations (the two parameters are independent)

d ,C 1= −ij represents a strong correlation but with an opposite effect (if one increases, the other decreases). The objective function which is used in this work is:

( ) ( ) ( )*P PE M Mα α α= − … … … … … … … … (2.7)

where, • α is the vector of unknown reservoir model parameters • ( )PM α is the vectors of simulated reservoir production histories

• ( )*PM α is the vectors of observed production historical data

In our study, the vector of unknown reservoir model parameters includes porosity and permeability for all active cells. ( )*

PM α vector includes well bottom hole pressures for both injectors and producers (WBHPI and WBHPP, respectively), and well oil and water production rates (WOPR and WWPR, respectively). No prior information in the objective function is considered in this work. 2.7.3. Optimization method In mathematical formulation, optimization is the search of maximum and minimum in the value of a certain response function done in an iterative way. The starting point can be one or several points picked randomly or carefully. Different criteria may be used to stop the iteration: maximum number of iterations or when no further improvement is expected. In history matching, the objective function will never reach the value zero, especially when the prior term is included (Harb, 2004). The general form of optimization problem shows in the following equation:

n

min f ( x ) { c ( x ) 0,subject to

x R { c ( x ) 0, I

ε= ∈

∈ ≥ ∈

i

i

i

i… … … … … … (2.8)

where, • ci is the constraint function • I is the identity matrix

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The aim of optimization is to find the optimum condition for each parameter to have a minimum objective function *E( )α =

r min E( )αr .

*αr will be global minimum [Landa, (1979)] when

*f ( ) f ( )α α≤r r for all Ω∀∈ … … … … … … … (2.9)

*αr will be local minimum when there is a neighborhood N of *αr such that

*f ( ) f ( )α α≤r r for all N∀∈ … … … … … … … (2.10)

For the fastest optimization algorithms, local solutions with smaller point of objective function comparing with the neighboring points are needed. Global solutions are useful for some cases, but they are difficult to identify and locate. Optimization problems are classified as:

Nature of objective function and constrains: linear, nonlinear, convex, Existence and non-existence of constrains: unconstrained and constrained

optimization, Smoothness of the function: differentiable, non-differentiable.

For unconstrained and smooth functions optimization condition the equation will be:

* EE( ) 0αα∂

∇ = =∂

rr and Tx Hx 0; x 0> ∀ ≠

r r r … … … … (2.11)

where, • E is the objective function • α is the reservoir parameter • H is the Hessian matrix

2EH

α α∂

=∂ ∂i i

… … … … … … … … … … … (2.12)

1,1 1,npar

npar ,1 npar ,npar

H ... HH . .

H ... H

⎡ ⎤⎢ ⎥=⎢ ⎥⎣ ⎦

… … … … … ... … . (2.13)

H is positive and is second derivative of objective function ( E ) to the parameter α . Using second order Taylor expansion the equation will be:

T1E( ) E( ) E H2

α Δα α Δα Δα Δα+ = +∇ +r r r r r r … … … … (2.14)

when *( )α α→r r , then *( E E )∇ →∇ and *( H H )→

17

* * T1E( ) E( ) H2

α Δα α Δα Δα+ − ≈r r r r r… … … … … .. (2.15)

*H is defined positive if *αr is a minimum. Foss (1987) proposed a method for calculating the uncertainty in the parameters by using the Hessian of the objective function which is a complex task for large objective function in large reservoir models. 2.8. Quantifying uncertainty in production forecasts It is widely recognized that the future production performance of oil and gas reservoirs cannot be predicted exactly. It is always challenging to reduce and quantify the uncertainty. Nowadays, more and more effort is being made to quantify this uncertainty. One of the aims of this work is to quantify uncertainty in future reservoir performance. In particular, it considers reservoirs where some production data (beyond well testing) is available. Such data is particularly difficult to incorporate in an uncertainty analysis because of the time consuming nature of the computations necessary to simulate fluid flow in the reservoir. For reservoir management, future forecasting is a very necessary step where the following procedures are common:

Construct an initial reservoir model, Adjust this model until the simulated production data matches the historical

production data, Use this adjusted model to simulate the future production.

2.8.1. Errors induced Saleri (1993) describes four sources of uncertainty in the forecasting of a reservoir performance: • Model Characterisation: When running history matching model, reservoir

geometry is fixed while a limited amount of parameters varied. The model errors depend on the choice of numerical simulator, i.e., mass balance or streamline, finite element or finite difference. These errors arise from the approximation of the originally continuous conservation and flow equations with discrete analogues, and the inability to capture sub-grid details. Consideration must be given to errors due to numerical diffusion and cell aspect ratio (Subbey, et al. 2003).

• Scale-up: All models (geostatistical and reservoir) are up-scaled for handling in a

reasonable time by the simulator. The main drawback in up-scaling is a loss of information in the cell properties that are considered as homogeneous. In reality, to

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reproduce the heterogeneities present in the reservoir, it is always desirable to have the flow simulator grid as fine as possible.

• Mathematical: The mathematical error occurs while representing the real numbers

in the numerical computation, which can be minimized but not eradicated (Gill, et.al. 1984). Also exactly correct models do not always give exact reality due to the discretization of the flow equation by the finite-difference method introduces automatically errors in the process.

• Data Quality: Consistent input data is vital for the success of a simulation study.

Errors in input data will inevitably lead to errors in the final results (Fonseca, et al. 2004). Also production data smoothing is an important step.

2.8.2. Sensitivities The sensitivity coefficients are the partial derivatives of the reservoir variables (such as pressure and saturations) with respect to the reservoir parameters (such as porosity and permeability) which can be computed from the sensitivity of the pressure and saturation:

calc

,( )s Δ

Δ= i

i jj

y xx

… … … … … … … … … … (2.16)

The calculation of the sensitivity coefficients can be done by perturbation method. In this method, every parameter is slightly perturbing with respect to time and running the flow simulator to observe the variation of responses. For example: in the Ensemble Kalman Filter, there are N parameters in the model and this method requires (N+1) forward simulations. 2.8.3. Selection of meaningful parameters In a reservoir model, there are a large number of uncertain parameters present that increases the minimization inefficient and is more difficult. For the updating of the reservoir parameters need the selection of most sensitive and effective ones. According to Lepine, et al. (1998), this selection is based on two factors:

The sensitivity of the results to the parameters, The range of uncertainty in the value of parameter.

Thus an automatic or semi-automatic method is necessary for selection of the crucial parameters that would be easier and faster. In this study, Ensemble Kalman Filter has chosen and is described in Chapter 4. During the production life of the reservoir, collected data are divided into two major groups: static and dynamic. Static data are not related to movement of fluid in the reservoir such as: geology data, electrical logs, core analysis, fluid properties, seismic and

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geostatistics whereas dynamic data are related to the movement of fluid in the reservoir such as: well testing data, pressure shut in survey, production history, bottom hole pressure from permanent gauge, water cut, gas oil ratio etc. Permeability and porosity have a largest influence in determining the reservoir performance. One alternative approach to uncertainty prediction is the use of response surfaces, experimental design methods and Monte Carlo simulation (Lepine, et al., 1998). In this approach, the prediction quantity of interest, d, is a simple function of the input parameters, α. The function used is usually a polynomial of first or second order. A number of simulations are run in which the parameters α is perturbed about their base case values (often an ‘optimistic’ and a ‘pessimistic’ value are retained for each parameter in addition to the base case value). Then, regression method is used to find the coefficients in the response function that give the best fit to the results from these simulations. Experimental design techniques use to determine exactly which combination of parameter values to use in the simulations. Finally, Monte Carlo simulation is used to construct a probability distribution for the prediction quantity d. This technique is potentially more accurate because it makes no assumption of linearity in the simulator response. On the other hand, only the quadratic terms are usually considered in practice. The technique is also more costly, as the number of predictive simulations that have to run is at least as great as the number of coefficients retained in the response function. 2.9. Methods for uncertainty analysis Generating multiple initial simulation models (manual uncertainty analysis) is very costly and time consuming. Oliver, et al. (1996) has presented a refinement of this approach which estimates probability density of the parameters by using a combination of stochastic reservoir realization and automatic history matching with uncertainty. Automatic uncertainty analysis is another method for estimating uncertainties without using multiple initial reservoir simulations. These methods divided into two groups (Harb, 2004):

Those that add roughness to a smooth estimate (like the linearization methods), Methods that add a smooth correction to rough estimates (Like the Bayesian

stochastic methods). 2.9.1. Linearization method For estimating uncertainty in the future performance, the values of the parameters are perturbated slightly. In this case, to obtain an acceptable history match from the adjusted values of parameters, it will probably be possible to perturb these values slightly and still have a match that would be considered acceptable (Lepine, et al. 1998). From the theoretical point of view, if amount of uncertainty in parameter x was x±Δ , the amount of uncertainty in predicted quantity will be:

20

yy xx∂

Δ = Δ∂

… … … … … … … … … … … … … (2.17)

where, • y is the predicted quantity which can be bottom hole pressure, water and gas oil

ratio, oil production rate and etc. of some particular wells in some period of time • x is the input parameter which can be permeability, porosity of a reservoir

Reservoir simulator can calculate the reservoir simulator sensitivity yx∂∂

at relatively little

extra cost. Estimating uncertainty from single parameter in one predicted quantity is not realistic, but it will be more authenticated to estimate uncertainty in several predicted quantities that comes from several parameters. In this case these parameters may not be independent. 2.9.2. Bayesian approach This is a probabilistic method where reservoir parameters are considered as stochastic parameters and reservoir model is described with probability density function, pdf. The main objective is computing of Maximum a Posteriori, MAP, that often called analysis, estimates of reservoir characteristic and forecast of future production. From Bayes’ theory a posteriori pdf is divided into two parts:

a) Prior pdf, b) Likelihood model.

obs obs

prior priorf ( x / x , y ) f ( y / x ) f ( x / x )∞ × … … … … … (2.18)

a) Prior pdf, priorf ( x / x ) represents the amount of uncertainty in the reservoir is often called the forecast in geosciences. Prior stochastic function is set up with some well known rock physical relationship. The model assumptions are based on available reservoir knowledge and general rock physical relationships that show the relationship between relative changes in the parameters. These relationships give expectation values for the prior distributions. In addition, all uncertainties in rock physics relationship and reservoir properties will be added as a Gaussian error to all expectation values with mean value of errors to zero and variance 2

ασ . In a Gaussian approximation, the prior pdf is a Gauss

prior prior( x ,C ) .

T 1prior prior prior prior

1f ( x / x ) c exp ( x x ) C ( x x )2

−⎛ ⎞= − − −⎜ ⎟⎝ ⎠

… … …(2.19)

where, • c is the normalization constant

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b) Likelihood model, obsf ( y / x ) represents how likely it is that the model parameter x could give the observation obsy , i.e. the difference between the predicted data and the observed data in a given model. Moreover, it describes how measured changes in parameters are related to the relative changes in observation data. The posterior pdf can be evaluated from:

1T cal obs 1 cal obs

prior prior pprior1 1( x ) c exp ( x x ) C ( x x ) ( y ( x ) y )C ( y ( x ) y )2 2

π − −⎛ ⎞= − − − − − −⎜ ⎟⎝ ⎠

… … … … … … … … .(2.20) A synthetic reservoir model Case 1 and a semi-synthetic reservoir model from Norne field are created to observe the production forecast. From the first case, ten years production data and from Norne field, twenty years production data has been considered. Those data were simulated using a commercial reservoir simulator (ECLIPSE 100) for History Matching together with quantifying the uncertainty associated with their forecast.

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Chapter 3 Statistic Fundamentals Some basic statistic principles are needed to understand the Ensemble Kalman Filter (EnKF). The statistics given in this chapter are a short overview of some of the fundamentals, providing a basis for understanding the filter theory. 3.1. Random variables Random Variable (RV) is a variable which gives a series of outcomes, each with a certain probability or frequency of occurrence. It is denoted with the capital letter X and the possible outcomes are denoted with the corresponding small letters ,i{ x i = 1, ... ...n} , for a discrete variables with n outcomes, or min max{ x [ x , x ]}∈ for a continuous variable valued in the interval bounded by a maximum and minimum value. A real random variable is a real finite-valued function .X ( )defined on Ω if, for every real number, the inequality

( )X x≤w … … … … … … … … … … … ..(3.1)

defines a set w whose probability is defined. The function

( ) ( ){ }XF x Pr X x [0,1]≤ ∈w … … … … … … ..(3.2)

is called the cumulative distribution function (CDF) where A cumulative distribution function (cdf) providing the probability for the random variable (RV) not to exceed a given threshold value x .

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3.2. Probability density function A probability density function (pdf) is defined as the derivative of the cumulative distribution function (cdf) at x -values of non-discontinuity. It states the probability that a random variable X will take a particular value x. The PDF ( )Xf x must satisfy the Equation (3.3a) and (3.3b).

( )Xf x 0 x≥ ∀ … … … … … … … … … … (3.3a)

( )Xf x dx 1∞

−∞=∫ … … … … … … … … … … (3.3b)

3.3. Expected value, Variance and Covariance 3.3.1. Expected value The expected value of a random variable is the integral of the random variable with respect to its probability measure. For discrete random variables, this is equivalent to the probability -weighted sum of the possible values, and for continuous random variables with a density function, it is the probability density -weighted integral of the possible values. The expected value for a random variable X is given both in the discrete and the continuous case by Discrete Continuous

E[X] =∑=

n

1iii xp E[X] = ( )Xx f x

∞

−∞∫ … … … … … (3.4)

where, • ( )Xf x is the pdf for the continuous case • pi is the probability of X = xi for the discrete case

Given a sample mean where the expected value that is the expected outcome if infinitely many data are present can be approximated by

[ ]N

ii 1

1E X x xN =

= ∑ … … … … … … … … … (3.5)

where, • N is the independent realizations of x • x has dimension m and the distribution f(x)

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3.3.2. Variance The variance of a random variable is a measure of statistical dispersion, averaging the squared distance of its possible values from the expected value (mean). Let X be a numerically valued random variable with expected value E(X). Then the variance of X, denoted by VAR(X) is

VAR (X) = E [(X − E[X])2] = E[X2] − E[X]2 … ... ... ... (3.6)

In a qualitative sense, the variance of iX is a measure of the dispersion of X about its mean. Another property of the random variable X is the standard deviation (σ), which is defined as the square root of the variance. The sample variance can be approximated as shown in Equation (3.7)

[ ] [ ]( ) ( ) ( )N 22 2

ii 1

1VAR X E X E X x x x xN 1 =

⎡ ⎤= − − = −⎢ ⎥⎣ ⎦ − ∑ … … (3.7)

where, • N − 1 is used rather than N to provide an unbiased estimator for the variance

3.3.3. Covariance Covariance is a measure of how much two variables change together. The covariance is defined in Equation (3.8) for two random variables X and Y.

COV [X,Y] = E [(X − E[X]) (Y − E[Y])] … … … … ..(3.8)

Here the joint probability function can be defined as f(X, Y). If X and Y are independent f(X, Y) = f(X) f(Y) and accordingly COV [X, Y] = 0. Qualitatively the covariance describes the dependency between the two random variables X and Y. The sample covariance is derived using the same properties as of sample variance above and is given by

[ ] [ ]( ) [ ]( )COV X ,Y E X E X Y E Y⎡ ⎤= − −⎣ ⎦… … … … … … (3.9)

( )( ) ( )( )N

i ii 1

1x x y y x x y yN 1 =

− − = − −− ∑ … … …(3.10)

3.4. Probability distributions Probability density function (pdf) has given a certain distribution for random variables. The most common is the normal or Gaussian distribution. A Gaussian function (named

25

after Carl Friedrich Gauss) is the probability function of the normal distribution in the form of

2 2( x ) / 2X

1f ( x ) e2

μ σ

σ π− −= … … … … … … … …(3.11)

where, • e 2.718281828≈ (Euler’s number) • E[ X ]μ =

X ~ N (E[X], σ2) is short for telling that a random variable X is normal distributed with an expected value of E[X] and standard deviation σ. If X is a vector of Gaussian distributed random variables with mean E[X] and a covariance matrix Q, then this can be denoted X ~ N (E[X], Q). The diagonal elements of Q denote the variance for each of the random variables in X and the off-diagonal elements represent the covariance between the variables. 3.5. Approximations using samples from a distribution Above is an approximation for the expected value, variance and covariance that is based on using an ensemble of samples to describe a probability distribution. The theory used is based on the description given in (Liu, 2001) and (Robert and Casella, 1999). A description of an integral approximation using a Markov Chain Monte Carlo method is given in the following: Markov Chain Monte Carlo (MCMC) method: To derive the expected value for understanding the need of using approximations, an integral of variance and covariance is observed given in Eq. (3.12) which is an essential part of many scientific problems.

I = ∫D dx)x(g … … … … … … … … … … .. (3.12)

where, • D is a region defined in a high-dimensional space • g(x) is the function of interest

Solving this integral using numerical integration can become very complex at dimensions higher than 3-4. To cope with the problems of higher magnitude, an alternative calculation of the integral is available. Assume that N is independent and identically distributed, random samples x(1), . . . , x(N) are drawn uniformly from D. Then the following approximation of I can be obtained

26

( )( ) ( )( )1 NN

1I g x ...... g xN⎡ ⎤= + +⎣ ⎦ … … … … ... … .(3.13)

The average of many independent random variables with common mean and finite variances tends to stabilize at their common mean. This translates into

NNˆlim I I

→∞= with probability 1… … … … … … … (3.14)

The convergence rate is defined from the central limit theorem as follows

( ) ( )2N

ˆN I I N 0,σ− → … … … … … … … … (3.15)

where, ( )2 VAR g xσ ⎡ ⎤= ⎣ ⎦ . The "error term" of the Monte Carlo approximation is O(N−1/2). The error term is independent of the dimensionality of x, but two intrinsic difficulties arise

When the region D is large in high-dimensional space, the variance 2σ can be formidably large,

One may not be able to produce uniform random samples in an arbitrary region D.

Importance sampling is needed to compensate with this idea. The Markov Chain Monte Carlo (MCMC) method proposed here can be used to approximate the expected value, the variance and the covariance for a given sample.

27

Chapter 4 Theoretical Formulation of the EnKF For linear modeling, the original Kalman filter (KF) was designed and was first presented by R. E. Kalman in 1960. The Extended Kalman Filter (EKF), based on linearization of the nonlinear model using the Jacobian, was used as an early attempt to adapt this original Kalman filter (KF) to nonlinear problems which is not suitable for large scale problems or problems that are too nonlinear. The Ensemble Kalman Filter (EnKF) was introduced by G. Evensen in 1964 to handle the large nonlinear oceanic models which give promising results in many areas. The original Kalman filter and the Extended Kalman filter have the drawback of taking too much computational time and handling of nonlinear dynamics as well as computing error covariance matrix. In the Ensemble Kalman filter, the covariance matrix estimate (P) is predicted and analyzed using the ensemble statistics. 4.1. Linear Kalman filter The Kalman Filter (KF) is a good basis for understanding the EnKF. A good description of the Kalman filter is given in the book of (Brown and Hwang, 1997). 4.1.1. Problem formulation A linear dynamical model in a discrete stochastic system can be written in the following form shown in Equation (4.1),

k 1 k k kx x u qΦ+ = + Δ + … … … … … ... … … … (4.1a) yk = Cxk + vk … … … … … … … … … … . (4.1b)

28

where, • Φ is the linear model matrix • ∆ is the input matrix • uk is the inputs at time k • yk is the measurements at time k • C is the linear measurement matrix • xk is the true state at time k

The true initial state is given by x0 and it is normal distributed

N(0, P0)… … … … … … … … … … … … (4.2)

where, • P0 is the initial covariance matrix for the stochastic process x.

qk ~ N(0, Qk)… … … … … … … … … … … (4.3a) vk ~ N(0, Vk) … … … … … … … … … … … (4.3b)

where, • qk is the unknown model error at time k • vk is the measurement error at time k • Qk is the covariance matrices for the model noise • Vk is the measurement noise

These errors are assumed normal distributed as shown in Equation (4.3). 4.1.2. Kalman filter equations Considering the problem from Equation (4.1) where xk denotes the system state at time tk and xk+1 is state at some time tk+1, and tk+1 > tk. Let k 1x−

+ be the a priori estimate of the process state based on the previous information available until this point (tk+1). Similarly let kx be the best estimate of the system state with filtered information up to time tk. This can be written in mathematical terms as

( )k 1 k 1x E x y− ∗+ += | … … … … … … … … … .. (4.4)

( )k kx E x y∗= | … … … … … … … … … … (4.5) where,

• y* are all the available measurements up to time tk The model for a priori prediction of the states forward in time can be derived using Equation (4.1) is

k 1 k kˆ ˆx x uΦ−+ = + Δ … … … … … … … … … … (4.6)

29

The a priori covariance matrix is updated using the following matrix relationship

Tk 1 k kP P QΦ Φ−+ = + … … … … … … … … … .. (4.7)

where,

( )( )Tk 1 k 1 k 1 k 1 k 1ˆ ˆP E x x x x y− − − ∗+ + + + +

⎡ ⎤= − − |⎢ ⎥⎣ ⎦… … … … …(4.8)

( )( )Tk k k k kˆ ˆP E x x x x y∗⎡ ⎤= − − |⎣ ⎦ … … … … … … …(4.9) The filtering step is where the a posteriori estimates are calculated. The a posteriori state is updated using the following equation

( )k k k k kˆ ˆx x K y Cx− −= + − … … … … … … … … (4.10) where the Kalman gain Kk is given by

( ) 1T Tk k k kK P C CP C V

−− −= + … … … … … … … . (4.11) The a posteriori covariance matrices are updated as follows

( )k k kP I K C P−= − … … … … … … … … … (4.12)

The filter can be summarized in Algorithm 1.

input: 0x , P0, Vk, Qk k = 0 while true do Prediction step k 1 k kˆ ˆx x uΦ−

+ ← + Δ // Forecast the state ahead T

k 1 k kP P QΦ Φ−+ ← + // Forecast the covariance matrix ahead

Filtering step ( ) 1T T

k k k kK P C CP C V−− −← + // Compute the Kalman gain

( )k k k k kˆ ˆx x K y Cx− −← + − // Update state estimate with measurement

( )k k kP I K C P−← − // Update the covariance matrix k k 1← + end Algorithm 1: Basic Linear Kalman Filter (KF) (Jensen, 2007) Here, the two steps of prediction and filtering are illustrated. For the Kalman filter the initial estimate for the state ( 0x ) and the state covariance (P0) is needed together with the model error and measurement covariance (Qk and Vk respectively). These variables are used to tune the Kalman filter response. If the initial state and/or the state covariance are

30

close to its true value, then the settling time will naturally be shorter. Both the measurement and the model error covariance tuning adjust the Kalman gain Kk. Kalman gain decides how much the variables are updated from the information in the measurements. When the measurement covariance is large, this suggests that there is a lot of error in the measurement. As a result the Kalman gain will be smaller and the observed effect of the analysis step will decrease. If the model error covariance is large, the Kalman gain will increase and push toward a stronger influence from the analysis step. These two tuning parameters have to be considered together and in the end it is the ratio between them. 4.2. Ensemble Kalman Filter The basic idea behind the Ensemble Kalman Filter (EnKF) is to provide a filter used for large scale nonlinear systems. The Ensemble Kalman Filter is a Monte Carlo method where an ensemble of reservoir model is used. The EnKF runs multiple simulation models independently, assimilates the new measurements and updates the multiple models simultaneously. After each updating it describes mean and variance where mean represents the most probable model and variance represents the change range or uncertainty. The correlation between reservoir response (e.g., water cut and rate) and reservoir variables (e.g., permeability and porosity) can be estimated from the ensemble. An estimate of uncertainty in future reservoir performance can also be obtained from the ensemble. (Yaqing & Oliver, 2005). 4.2.1. Problem formulation The Ensemble Kalman Filter (EnKF) is used for large scale problems that are often arranged in a grid, where each grid block contains a number of static variables (porosity, permeability) and dynamic variables (pressure, saturation) which are a function of space and time. The model state can be defined as ( ) x

knx z,t ∈R and consists of nx dynamic

variables at each location z in space and at time tk. Some of the static variables are usually poorly known. Let these poorly known parameters be denoted ( ) nz θθ ∈R consisting of nθ static variables and is defined in space as the dynamic parameters. Consider the following problem

( ) ( ) ( )( ) ( )k 1 k kx z,t f x z,t , z q z,tθ+ = + … … … … … (4.13a)

( ) ( )0 z zθ = θ θ ′+ … … … … … … … … … . (4.13b)

( ) ( )( ) ( )k ky g x z,t , z v tθ= + … … … … … … … (4.13c) where,

• ( ) ( )( )kf x z,t , zθ is the nonlinear model operator

• q(z, t) is the model noise • q(z, tk) represents the model errors

31

The model state is initially given as x(z, t0). An initial guess of the poorly known parameters is given by ( )0

nx θθ ∈R . The measurements yny∈R may be direct point measurements of the solution or complex parameters nonlinearly related to the model state. Without the error term q(z, tk), the system would be overly specified and have no solution. But introducing these errors implies that there are infinitely many solutions to the system. To cope with the infinite possibilities an assumption is made where the errors are normal distributed with expected values being zero and the covariance known. 4.2.2. Ensemble covariance In general the error covariance matrices for the predicted and filtered ensemble are given by

( )( )Tk k k k kˆ ˆP x x x x− − −= − − … … … … … … … … (4.14)

( )( )Tk k k k kˆ ˆP x x x x= − − … … … … … … … … (4.15)

Since the true state is in general hard to acquire one can instead redefine the ensemble

covariance in regards to the ensemble means ( )k kˆ ˆx ,x− as shown in Equations (4.16 & 4.17).

( )( )Tk k k k kˆ ˆ ˆ ˆP x x x x− − − − −= − − … … … … … … … … (4.16)

( )( )Tk k k k kˆ ˆ ˆ ˆP x x x x= − − … … … … … … … … . (4.17)

The interpretation of introducing this enforces is that the ensemble mean is the best estimate and the spreading of the ensemble around defines the error in the ensemble mean. Now the error covariance can be represented using an appropriate ensemble of model states. Consider the error term from the Monte Carlo integration given in Section 3.5. The description of the error term also holds for the ensemble covariance. As the size of the ensemble N increases, the errors in the Monte Carlo sampling will decrease proportional to N1 | . Evensen (2007) concluded that the information contained by a full probability density function can be exactly represented by an infinite ensemble of model states. 4.2.3. Ensemble representation: Earlier the system states x and poorly known parameters θ were defined. These can be put

together in a matrix ( )n X N

kA z,t ∈R , where xn n nθ= + , holding the N ensemble members at time tk. In mathematical terms this becomes

32

( )( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )1 2 N

k k k1 2 Nk

k k k

x z,t x z,t ... x z,tA z,tz,t z,t ... z,tθ θ θ

⎡ ⎤= ⎢ ⎥⎣ ⎦

… … … … (4.18)

The ensemble mean can now be written as

( ) ( )k k NA z,t A z,t .1= … … … … … … … … … … .. (4.19)

where, 1N is a vector consisting of N × 1 elements of value 1/N. The ensemble covariance in space can then be estimated as

( ) ( ) ( )( ) ( ) ( )( )T1 2 k 1 k 1 k 2 k 2 k1P z ,z ,t A z ,t A z ,t A z ,t A z ,t

N 1= − −

−… … ..(4.20)

The measurements can also be formulated using an ensemble representation. For a system consisting of J measurements, J vectors consisting of N perturbed measurements is defined as

l t lj j jv , l 1,... ...,N j 1,... ...,J= + = =y y … … … … … … (4.21)

where t

jy is the "true" measurement from the original process. These variables can be stored in a measurement matrix jY and a measurement perturbation matrix jε as in

Equations (4.22) and (4.23) for each of the system measurements [ ]J 1...... j∈ .

( )1 2 Nj j j j, ,....=Y y y y … … … … … … … … … … … (4.22)

( )1 2 Nj j j jv ,v ,....,vε = … … … … … … … … … … … (4.23)

For the [ ]J 1...... j∈ measurement the covariance can be estimated using the ensemble measurement perturbations as follows

( )T

j jk j

V tN 1ε ε

=−

… … … … … … … … … … … (4.24)

4.2.4. EnKF algorithm The ensemble can be expressed as a function of time only by stacking the states and parameters. Given a finite set of space parameters z = [z1 . . . zg], S states and P poorly known parameters at each location. Each state ix , i∈ [1, S] and parameter jθ , j∈ [1, P] can then be expressed for all locations in space as follows:

33

( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

1 2 Ni 1 k i 1 k i 1 k

i k

1 2 Ni g k i g k i g k

x z ,t x z ,t ... x z ,t. . . .

t . . . .. . . .

x z ,t x z ,t ... x z ,t

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

X … … … … (4.25a)

( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 2 Nj 1 k j 1 k j 1 k

j k

1 2 Nj g k j g k j g k

z ,t z ,t ... z ,t. . . .

t . . . .. . . .z ,t z ,t ... z ,t

θ θ θ

θ

θ θ θ

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

… … … … (4.25b)

The A matrix can now be redefined from Equation (4.18) containing all ensemble states and parameters for all locations in space. This matrix works as an augmented state vector to be used in the Ensemble Kalman Filter. Combining Equations (4.25a) and (4.25b), the ensemble matrix can be derived as:

( )

( )( )

( )( )( )

( )

1 k

2 k

S kk k

1 k

1 k

P k

tt

tA A ttt

t

θθ

θ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥= = ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

M

M

XX

X … … … … … … … … ..(4.26)

The same notation can be used for the measurements using Equation (4.22).

( )( )( )

( )

1 k

2 kk k

J k

ttY Y t

t

⎡ ⎤⎢ ⎥

= = ⎢ ⎥⎢ ⎥⎣ ⎦

M

YY

Y

… … … … … … … … … (4.27)

The matrix containing the measurements from the true system is stored in Tt t t t

k 1 2 JY ....⎡ ⎤= ⎣ ⎦y y y . Similarly this can be done for the ensemble measurement perturbation using the Equation (4.23).

( )( )( )

( )

1 k

2 kk k

J k

tt

E E t

t

εε

ε

⎡ ⎤⎢ ⎥

= = ⎢ ⎥⎢ ⎥⎣ ⎦

M… … … … … … … … …(4.28)

For a general nonlinear system the measurement is given according to Equation (4.13c). The estimated states and parameters for all the ensembles are denoted kA . Let the measurements of the estimated ensemble at time tk be given by the following nonlinear relationship

34

( ) ( ) ( ) ( )( ) ( )( ) ( )( ) ( )( )1 N1 N1 Nk k k k k

ˆ ˆ ˆˆ ˆ ˆ ˆA .... g x , ....g x ,θ θ− − − −⎡ ⎤⎛ ⎞ ⎛ ⎞⎡ ⎤= = ⎜ ⎟ ⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠ ⎝ ⎠⎣ ⎦M y y … .. (4.29)

For convenience the estimate for the states, parameters and measurements are stacked in a large ensemble matrix as shown below

( ) ( )k

k kk

Aˆ ˆ t A⎡ ⎤

= = ⎢ ⎥⎢ ⎥⎣ ⎦

A A M … … … … … … … … (4.30)

The Ensemble Kalman filter updates the estimate of the states and variables by updating

the ensemble matrix 0A . Note that ( )AM , in the ensemble matrix from Equation (4.30),

is treated as a diagnostic variable in the system. This is because its update is not used in the next step of the filter. The EnKF also updates the mean of the ensemble matrix kA . The superscript −, as defined previously, denotes that an estimate is a priori. To initialize the filter, the model and measurement covariance Qk and Vk respectively must be provided. In addition an initial ensemble must be specified. Let Hk be the measurement index matrix at each time step k defined such that

( )k k kˆ ˆH A=A M … … … … … … … … … … (4.31)

Using this notation the Ensemble Kalman Filter is summarized in Algorithm 2. There are three sub functions, which involve the statistics in the presented algorithm.

input Qk, Vk

0 ⇐A computeInitialEnsemble while true do Prediction qd ← computeModelNoise (Qk)

kˆ −A ← ( )k 1

ˆf dq− +A

kP− ← ( )( )Tk k k kˆ ˆ ˆ ˆ− − − −− −A A A A

Filtering Ek ← computeMeasurementNoise (Vk) Yk ← t

k kY E+

Kk ← ( ) 1Tk k k k kP H H P H V

−− − +

kA ← ( )k k k k kˆ ˆK Y H− −+ −A A

kA ← ( )k k k k kˆ ˆK Y H− −+ −A A

kP ← ( )( )Tk k k kˆ ˆ ˆ ˆ− −A A A A

Algorithm 2: Ensemble Kalman Filter (EnKF) (Jensen, 2007)

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4.3. Theoretical comparison of linear Kalman filter (KF) and EnKF It is natural to examine the covariance matrices while comparing linear Kalman filter (KF) and Ensemble Kalman Filter (EnKF). Equation (4.17) is used for the definitions of the analysis scheme in KF. Burgers, et al. (1998) showed that it is essential that the observations are treated as random variables with a mean equal to the system observation and covariance equal to Vtrue. The ensemble of observations is given in Equation (4.21). Comparing the filtering step in the KF and the EnKF, the relation between the a priori and a posteriori ensemble is identical, apart from the use of kP− and kP . Given an ensemble of size N and the a posteriori update of the state and the state mean from Algorithm 2, the following relationship can be derived for ensemble member l .

( )( ) ( ) ( )lllllˆ ˆ ˆ ˆx x I KH x x K− −− = − − + −y y … … … … (4.32)

where, • lx is the ensemble matrix • lx is the ensemble mean • I is the integral • K is the Kalman gain • H is the measurement index matrix • x− is the a priori estimate • ly is the true measurement at space l

The Kalman gain is the same as in the formula from both Algorithms 1 and 2. The following derivation can be made using this relationship and considering the ensemble covariance from the EnKF

( )( )( )( ) ( )( )

( )( )( ) ( ) ( )( )( ) ( )

( )( )

T

k k k k k

k k k k

T T T Tk k k k k k l

T Tk

T T T Tk k k k

k

ˆ ˆ ˆ ˆP x x x x

ˆ ˆI KH x x K ...

ˆ ˆ ˆ ˆx x x xI KH I KH K KI KH P I KH KVK

P KHP P H K K HP H V KI KH P

− −

− − − −

−

− − − −

−

= − −

= − − + −

− −= − − + − −

= − − += − − + += −

y y

y y y y… (4.33)

The result is the minimum error covariance as used in the KF scheme. The conclusion is that the EnKF analysis step converges to that of the KF when the ensemble size converges to infinity. To hold the observations, ( y ) must be treated as random variables.

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4.4. EnKF methodology Ensemble Kalman Filter is a procedure for automatic history matching that requires collaboration between many areas and has different steps:

Create an initial model of the reservoir based on geological interpretation, Parameterize the estimation problem (static and dynamic variables,

measurements, etc.), Define the problem in mathematical terms, Create the initial ensemble, Run the filter, Interpret the results.

Unsatisfactory results should be analyzed again from the beginning to find the error and redo the procedure from where it went wrong. 4.5. Managing the uncertainties EnKF is based on statistics. On the basis of this property, generation of ensembles or adding model/measurement noise is built which is important for the filter to work. The model noise describes model uncertainty in the filter that consequently comes in many parts of the EnKF algorithm. The characteristic of EnKF is to solve both the parameter and the state estimation problem. The adjustment of the model noise varies with the different problem cases and is based on the severances of the nonlinearities of the problem. The essential part of the Kalman filter is to construct the initial ensemble. The initial ensemble contains information about the initial states, parameters and their uncertainties. There are various strategies for creating these samples representing the uncertainties. Evensen (2007) suggested three sampling strategies that might be applied:

a) Sampling all variables randomly. The mean and variance will vary within the accuracy that can be expected for a given ensemble size,

b) Sampling all variables randomly, but correcting the sample such that it will have the correct mean and variance,

c) Using the improved sampling scheme presented by Evensen. From running several tests Evensen concluded that there was a slight improvement from strategy a) to strategy b). The third and more complex strategy gave a better result than the two previous even though it is not so computationally expensive. The improved sampling scheme seeks to generate ensembles with full rank and better conditioning than using a random sample. Evensen (2004) suggested that since the improved sampling scheme also generates a better conditioning of the ensemble during the forward integration it might be useful when computing the model noise. The improved sampling can be used in the following two ways:

37

Reduce the computation time by decreasing the ensemble size and still obtain the same results,

Improve the EnKF results even further. These sampling strategies have a common factor where samples are created using the mean and the covariance. Also the ensemble must be Gaussian distributed that means the probability density function (pdf) of the a priori ensemble is assumed to be Gaussian distributed. The Gaussian assumption result in EnKF solving for the mean and not for the mode. 4.6. Advantages and disadvantages of EnKF The main advantages of the Ensemble Kalman Filter is that it is simple and model independent, that it automatically filters out through nonlinear saturation processes such as convection that are fast and cannot be linearized, and that it provides optimal initial ensemble perturbations. It can be built upon any reservoir simulator as it requires only the output of the simulator. Thus the coding of EnKF is significantly simpler. Also if the number of ensemble member K is large enough, EnKF becomes more accurate compare to other Kalman Filters. It generates a set of updated reservoir models representing the uncertainty in the model. The Ensemble Kalman Filter is a Bayesian Approach which comes from the Bayes’ Theorem and is initialized by generating permeability and porosity fields using a priori geostatistical assumptions. The production data are incorporated sequentially with time, and the permeability and porosity fields are updated as new production data are introduced. Together with the static fields (permeability and porosity), the dynamic fields (reservoir pressure and saturation) are also being updated which is suitable for online updating of the model. The estimated fields depend on the initial ensemble which is generated stochastically. In this study, to get a reasonable estimate of the uncertainty of the model parameters, initial ensembles of size 100 were used to compare the results from the same distribution. The Ensemble Kalman Filter is appropriately designed to get production forecasts with uncertainty. A range of forecasts can be generated by running one forecast from each of the updated ensemble members. The EnKF method is computationally efficient and easy to implement compared to the traditional methods. In the updating of the dynamic fields, nothing is imposed to enforce the updated variables to be consistent with reservoir equations, and it is interesting to compare forecasts based on the last updated ensemble, with forecasts obtained by rerunning the complete simulation from time zero using the estimated permeability and porosity fields contained in the last ensemble. In addition, the EnKF is well-suited for parallel computation since the time evolution of ensemble reservoir models is completely independent.

38

The main disadvantage of EnKF is the low dimensionality of the ensemble (K~100), which introduces sampling errors in the estimation of the background error covariance B, especially at long distances. Ensemble Kalman Filters have adopted a localization of B (Houtekamer and Mitchell, 2001) by multiplying each term by a Gaussian shaped correlation that depends on the distance between points (Gaspari and Cohn, 1999), and this approach works well and is easy to implement on systems that assimilate observations sequentially (Tippett et al. 2003). 4.7. Use of EnKF for reservoir parameter estimation 4.7.1. Steps used for EnKF The creation and maintenance of a reservoir model is very difficult and complex and includes geological modeling, seismic characterization, reservoir simulation and model conditioning with all available data. Before running the ensemble Kalman filter the important task is to consider all aspects of reservoir geology, make a good model and then to run the EnKF. There are various steps involved with EnKF for estimation of reservoir parameters which are as follows:

a) To find parameters where the uncertainty lies, i.e., parameterization of fluid contact, porosity, permeability, pressure, saturation, transmissivity etc.,

b) Measurements in the facilities have also taken under consideration to tune the reservoir model,

c) To derive the ensemble Kalman filter by considering the state vector, d) To initialize the EnKF by searching initial ensemble of the parameters using

effective degrees of freedom by spanning smaller space, e) To reduce the problem by assuming the porosity and permeability to be smooth, f) To run EnKF, g) Interpreted and verified the results.

Regarding a) to g): For parameterization, the structural model is assumed to be accurate as the structural parameters cannot be estimated by using EnKF (Evensen, 2007). But uncertainty lies mainly in GOC, WOC etc. where initial information usually is derived from the drilling of wells through the specific areas that often leads to a large uncertainty in these important parameters. Porosity and permeability have the ability to transport fluids in the reservoir are the main parameters to adjust both the observed rates and the timing of water and/or gas breakthrough. The state parameters in the reservoir such as pressure P and the saturations of gas, oil and water (Sg, So and Sw) need also to be estimated.

39

With the static and dynamic variables, the measurements need to choose wisely as the measurements are the only information available to tune the model. But in general the measurements are given by facilities and there is no real choice. Once the parameters and states to be estimated are defined, an augmented state vector for the filter can be derived as

( ) ( )

( ) ( )( ) ( )( ) ( )

li 1 k

li g kl

j 1 k

lj g k

x z ,t

x z ,tz ,t

z ,t

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

M

M

q

q

l … … … … … … … … … (4.34)

where, • ( ) ( )l

i 0 kx z ,t denotes the states for ensemble l in grid block 0 at time k

• ( ) ( )lj 0 kz ,tq denotes the poorly known parameters to be estimated

4.7.2. Steps used for parameter estimation To include estimation of static parameters the state vector is extended (Stengel, 1994) and in theory the system can be described as follows:

TU [VPD ]= … … … … … … … … … … (4.35) where,

• V defines the dynamic state variables consists of pressure, water and gas saturations and solution gas/oil ratios for each grid block in the numerical solution scheme

• P is the model parameters of porosity and horizontal and vertical (logarithmic) permeability for each grid block

• D represents the measured production data and consists of bottom hole pressure, gas/oil ratio, and water cut for each well in the reservoir

The initial ensemble 0U is constructed on the basis of Gaussian Random Fields (GRFs) and depends upon the geostatistical assumptions about the reservoir. The permeability and porosity fields are generated stochastically based on the a priori geostatistical assumptions where the uncertainty of the dynamic variables was ignored in the initial state. Initially the dynamic state variables are defined by an equilibrium condition which is equal to the equilibrium state of the ‘true’ model.

40

One iteration of the ensemble Kalman filter consists of two steps, a forecast step fU and an analysis step Ua . The forecast step is calculated by using the model function (reservoir simulator) to propagate the state vectors (ensemble) to time step n-1 to time step n.

f, ,1U f (U )−= a

n i n i … … … … … … … … … … (4.36) where, i runs from 1 to the number of ensemble members N. In the analysis step, the forecast state vectors fU n are updated by taking into account the mismatch between measurements and the corresponding predictions from the ensemble members. The state vectors are related to the measured variables through the following equation

D HU= … … … … … … … … … … … … (4.37) where, H is a matrix that selects calculated measurements from the state vector. To get a linear relationship between U and D, it is necessary to include measured variables. Assume that the true observation vector at time n is given by

, ,o o oD D= +∈n n ni i … … … … … … … … … … (4.38)

where, each ,o∈n i is drawn from a normal distribution with zero mean and covariance

matrix Rn . The analyzed state is now computed as

, , , ,f o fU U K D HU )= + ( −n n n n ni i i i

a … … … … … … .. (4.39)

where, Kn is called the Kalman gain matrix and is given by

T T 1f fK P H ( HP H R )−= +n n n n … … … … … … .. (4.40)

The matrix fPn is an approximation to the model error covariance matrix and can be written as

Tf f fP L ( L )=n n n … … … … … … … … … … . (4.41)

where fLn is given by

,1 ,Nf f f f f

1 ˆ ˆL (U U )......(U U )N 1

⎡ ⎤= − −⎣ ⎦−n n n n n … ... … … (4.42)

here . represents ensemble mean. Assuming normal distributions, the mean of the analyzed ensemble

N,

1

1U UN =

= ∑n n ia a

i… … … … … … … … … … (4.43)

is the best estimate for the model state at time step n .

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The rate and pressure measurement which is used in the filter updating is also to be defined. Now the model is defined and the ensemble can be written in the form presented in Equation (4.30). After defining the problem, the next step is to initialize the EnKF by reducing the degrees of freedom by spanning a smaller space in which the filter searches for the solution. Consideration should be put into the problem reduction, so that the solution does not fall outside the span. Normally the problem reduction can be done by assuming the smoothness of the field’s permeability and porosity by using vertical and horizontal correlations. After the problem is parameterized and the ensemble is initialized, the EnKF is ready to run. The EnKF algorithm can be implemented with different options that might be turned on or off. These options are in general available to implement extra features that might improve the EnKF’s ability to solve the problem. 4.7.3. Measures of filter performance The filter performance is evaluated by using estimated parameter fields in history matching exercises. Estimated permeability and porosity from a given time step are used as initial values for the simulator, together with dynamic variables in the equilibrium state. Synthetic measurements are generated by running the simulator through the complete history matching period. The history match measure is given as:

T2M M o h,k

21 ,

D ( ) D ( )1HM( k )M σ=

⎡ ⎤−= ∑ ∑ ⎢ ⎥

⎢ ⎥⎣ ⎦

nn n

n j=1 n j

j j… … … … (4.44)

where, • M is the total number of measurements • MT is the number of the assimilation steps where the state vector is updated • Mn is the number of measurements at step n

• h,kDn are the vectors that are the calculated (history matched) measurements when fixed estimation static parameters estimated at timestep k are used

• h , the subscript is added to distinguish these values from the Kalman filter output • ,σn j is the standard deviation of the measurement error

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Chapter 5 SGSIM In this thesis work, the SGSIM (Sequential Gaussian Simulation) is used as the basis for achieving the result for Error Estimation and Cost Function by using simulation model. The data file and the initial ensemble of Case C of the Norne field is prepared using the Sequential Gaussian Simulation (SGSIM) to compare with the other two cases of Norne. SGeMS software is used for SGSIM which is the Stanford Geostatistical Modeling Software developed at Stanford University, USA. Several geostatistics algorithms, generally space-time distributed phenomena are used for modeling of earth system. It is user-friendly software used for Reservoir Forecasting with more recent geostatistics algorithms and recent multiple-point statistics simulation. 5.1. Fundamentals Error Estimation is the difference between an estimated value and the true value of a parameter or, sometimes, a value to be predicted. When a physical measurement is made, there is always some uncertainty about its accuracy. Every estimation method involves an estimation error, arising from the quantity to be estimated which generally differs from its estimator z , thus implying an error of estimation ˆz - z . Cost Function is a function of input prices and output quantity. Its value is the cost of making that output given those input prices. For example: 1 2c( w ,w , y ) is the cost of making output quantity y using inputs that cost 1w and 2w per unit. Geostatistical algorithms have been used to model the spatial distribution of uncertainty. The sample variogram is used in geostatistics to model the relationship between distance and variance; i.e., spatial autocorrelation which is a correlation of a variable with itself through space. Geostatistical algorithms can be divided into two broad categories:

• kriging • simulation

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Kriging produces a weighted linear estimate of the concentration at unsampled locations; the weights are derived from the sample variogram. Probability distributions of uncertainty estimated from kriging are based on two assumptions:

constant variance in concentration across the site (homoscedasticity), uncertainty distribution which can be modeled with a normal distribution.

Probability distributions for uncertainty estimated with geostatistical simulation algorithms are derived from multiple realizations that represent equi-probable estimates for the spatial distribution at unsampled locations. The distributions for uncertainty derived from simulation are indirectly dependent upon the above two assumptions. Geostatistical simulation relies on the definition of a stochastic model (e.g. a random field characterized by the set of its finite-dimensional distributions), a spatial domain and an algorithm used to construct realizations of the model over the domain. In practice, most algorithms are approximate, because their implementation requires simplifications or because the convergence to the model is only asymptotic. 5.2. Sequential simulation The main challenge of geostatistics is to make an easy solution from many unknowns and data of different types. This can be done by using the method called ‘divide and conquer’. The following are some models which will make the problem easier to solve:

To divide the problem into a series of easier problems where there is only one

unknown, uz( ) at a time. This will give the probability result of the unknown. Sequential simulation algorithm will help to recombine this elementary probability results,

To divide the large and complex data set n( u ) , which constituted of many different data types into a set of smaller more homogeneous data set kn ( u ) , where k = 1, … …, K. Then to combine those sets to a single set to get the result.

For the first model, three independent unknowns 1u )z( , 2u )z( and 3u )z( in three different locations u1, u2 and u3 will be considered. These three independent variables are also related with three different attributes. The joint probability density function (pdf) of three random variables conditional to the same data set (n) can be decomposed as (Goovaerts, 1997, p.376):

1 1 2 2 3 3Pr ob{ Z( u ) ,Z( u ) ,Z( u ) |( n )}= = = =z z z

1 1Pr ob{ Z( u ) |( n )}= z 2 2 1 1Pr ob{ Z( u ) |( n ),Z( u ) }= =z z … … … … … (5.1)

3 3 1 1 2 2Pr ob{ Z( u ) |( n ),Z( u ) ,Z( u ) }= = =z z z

44

The tri-variate joint pdf has been decomposed into the product of three univariate conditional pdf’s, each involving only one variable, 1Z( u ) first, then 2Z( u ) , last 3Z( u ) , but with increased data conditioning. The decomposition (Equation 8.1) then allows the process of sequential simulation, more precisely:

a value for 1Z( u ) is drawn from the first pdf 1 1Pr ob{ Z( u ) |( n )}= z , the simulated value is ( l )

1z next a value for 2Z( u ) is drawn from the second pdf

( l )2 2 1 1Pr ob{ Z( u ) |( n ),Z( u ) }= =z z , that value is ( l )

2z last a value for 3Z( u ) is drawn from the third pdf

( l ) ( l )3 3 1 1 2 2Pr ob{ Z( u ) |( n ),Z( u ) ,Z( u ) }= = =z z z , that value is ( l )

3z The three simulated values ( l )

1z , ( l )2z , ( l )

3z , although drawn sequence one after the other, stem from the joint tri-variate distribution conditional to the common data set (n) (Remy, et al. 2007). The problem of jointly simulating many variables will be possible by simulating one variable at a time but with an increasing conditioning data set, from (n) to (n+1) then (n+2). The problem created by increasing data set size is solved by retaining into the conditioning data set of each variable only the closest or most related previously simulated values (Goovaerts, 1997). 5.3. Estimating the local conditional distributions The critical step in sequential simulation consists of estimating the conditional distribution from a given specific conditioning data set n( u ) at each location u along the simulation path. There have been two avenues for the determination of the single variable Z( u ) conditional probability distribution function (pdf) Pr ob{ Z( u ) | n( u )}= z , both calling for a multiple-point (mp) Random Function (RF) model. The traditional 2-point statistics approach consists of evaluating the relation of the single unknown Z( u ) with one datum Z( u )α at a time. Such a relation typically takes the form of a covariance / correlation, or equivalently, a variogram; which are 2-point statistics. Then calling on a prior multiple-point model requiring only 2-point statistics for its calibration, the previous conditional pdf Pr ob{ Z( u ) | n( u )}= z is determined through some form of kriging. An example of such simple multiple-point (mp) model is present which can be calibrated from 2-point statistics are:

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the multivariate Gaussian model underlying the sequential Gaussian simulation algorithm (SGSIM),

the truncation at order two of an expansion of the exact conditional probability in the discrete case. Such truncation underlies the indicator simulation algorithm (SISIM).

In the above situation, the 2-point statistics approach aims at dividing the data set (n) into single locations or variables. First each single data variable is related to the single unknown (1+1=2-point statistics), then these elementary results are pieced together through some simple prior probabilistic model. Conditional probability is the probability of random variable X, given the occurrence of another random variable Y. Conditional probability is written P(Y | X ) , and is read "the probability of Y, given X" which is a probability distribution of Y when X is known to be a particular value. The conditional probability mass function of Y given X is

P(Y X p( )p( | ) P(Y | X

P( X ) p( )= =

= = = ==

y, x) x,yy x y x) =x x

… … .. (5.2)

and for continuous random variables, the conditional probability density function is

f ( )f ( | )f ( )

=x,yy xx

… … … … … … … … … … … (5.3)

In a single unsampled continuous variable at location u, denoted by data set n( u ) , the relevant cumulative distribution function (cdf) provides an assessment of the uncertainty about the unsampled value uz( ) that is specific to the location u and the data set n( u ) and is written

F( u; | n( u ) ) Pr ob{ Z( u ) | n( u )}= ≤z z … … … … (5.4)

In words, the probability that the unknown Z( u ) be valued is no greater than the threshold value z conditional to (knowing) the data set n( u ) . The conditional probability is, by definition, equal to the following ratio, with as numerator the probability of the event to be assessed Z( u ) ≤ z occurring jointly with the data event and as denominator the probability of that data even occurring:

( ) ( )( )

Pr ob{ Z( u ) n u }Pr ob{ Z( u ) | n u }

Pr ob{ n u }≤

≤ =z,

z … … … … (5.5)

Equation 5.5 makes explicit the dependence of that cdf on the location u, more precisely, the relation of that location with the n( u ) data retained. In all rigour, one should also

46

make explicit the type, location and value of each datum constituting the set n( u ) . Indeed if any aspect of that data set changes, the distribution of the equation changed. The above distribution is also called the Conditional Cumulative Distribution Function (ccdf) of a specific random variable Z( u ) given the data set n( u ) . There cannot be any probabilistic estimation or simulation without the necessary multiple-point (mp) statistics linking the data taken altogether to the unknown(s). Those mp statistics are either delivered explicitly through an analytical multivariate model or a training image or they are implicitly provided by the specific simulation algorithm retained. The role of an mp simulation is data conditioning. Multiple-point statistics simulation or MPS is a more recent modeling approach proposed by Guardiano and Srivastava (1993). MPS simulation is a reservoir facies modeling technique that uses conceptual geological models as 3D training images to generate geologically realistic reservoir models. The training images provide a conceptual description of the subsurface geological bodies, based on well log interpretation and general experience in reservoir architecture modeling. MPS simulations extract multiple-point patterns from the training image and anchor the patterns to the well data. A Random Function (RF) denoted by Z( u ) , is a set of dependent random variables

( ){ Z u , u S }∈ , each marked with a coordinate vector u, spanning a field or study area S. That field is typically a 3D physical volume, where u ( x, y,z )= is the vector of the 3 Cartesian coordinates; the common notation (z) for the variable and the vertical coordinate does not usually pose a problem. The variable could also be time in which case u t= , or it could involve both space and time as for atmospheric pressure in which case u ( x, y,z,t )= . Covariance and correlation are related parameters that indicate the extent to which two random variables co-vary. Variogram 2 ( x, y )γ is a function describing the degree of spatial dependence of a spatial random field or stochastic process Z( x ) . It is defined as the expected squared increment of the values between locations x and y (Wackernagel, 2003):

22 ( x, y ) E( Z( x ) Z( y ) )γ = − … … … … … … … (5.6)

The main tool for traditional geostatistics is the covariance or its equivalent or the variogram. Covariance and variogram are used to calculate variances and variances are non-negative. Computing experimental variogram and modeling them are the key steps of the traditional geostatistical studies. Fitting an analytical model to an experimental variogram achieves two purposes:

47

It allows computing a variogram value ( h )γ for any given lag vector h. Indeed, geostatistical estimation and simulation algorithms require knowledge of the variogram at arbitrary lags,

A model is a way to filter out the noise from the experimental variogram. Noise in the experimental variogram is typically a consequence of imperfect measurements or a lack of data.

The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. Each member of the family may be defined by two parameters, location and scale: the mean ("average", μ) and variance (standard deviation squared, σ2) respectively. The shape of all conditional distributions is Gaussian (normal) and the mean and variance are given by kriging. The standard normal distribution is the normal distribution with a mean of zero and a variance of one. Carl Friedrich Gauss became associated with this set of distributions when he analyzed astronomical data using them, and defined the equation of its probability density function. It is often called the bell curve because the graph of its probability density resembles a bell. The Gaussian Simulation is a powerful set of statistical and geostatistical tools specially tailored for uncertainty assessment and bivariate kriging. 5.4. Kriging Kriging is the source of the acceptance of geostatistics which is a major data integration tool. The kriging paradigm is the origin of most geostatistical algorithms whether aimed at estimation or simulation. Indicator kriging form with a single (normal) equation, it identifies Bayes’ relation and the very definition of a conditional probability (Journel, 1983). Kriging is in essence a generalized linear regression algorithm (Luenberger, 1969), extending the data-to-unknown correlation to data-to-data correlation through a non-diagonal kriging matrix. It is a regression with a non-independent data: actually it can be shown that kriging consists, first of de-correlating the data by defining linear combinations of the original data which are orthogonal for a given covariance/variogram model, then of a traditional linear regression from these ‘independent’ data transforms (Journel, 1989). Linear Regression Algorithm is the algorithm which is used as the well-known least squares method, where least squares can be interpreted as a method of fitting data. This algorithm determines the values of the model parameters by minimizing the quantity.

( )2 w y xα∑ −∑i i i j j ij … … … … … … … … … (5.7)

where, • α is the model parameters • i is the index that runs over all points in the dataset • wi is a weight factor for each data-point

48

The algorithm has no restrictions on the number of data-points, parameters or constraints. It has been implemented in such a way that it does not need any external routines, not even standard mathematical functions like "square root". There are several kinds of kriging, among them are: 5.4.1. Indicator kriging (IK) Indicator kriging is a kriging which is applied to variables that are binary indicators of occurrence of an event, say

{k1 if the event k occurs at location uI ( u ) 0 if not= … … … (5.8)

or for the continuous case:

{ kk

1 if Z( u )I( u; ) 0 if not≤= zz … … … … … ... … ..(5.9)

The event k to be estimated could be the presence of facies of type k at location u, or could be that the unsampled continuous variable uz( ) is valued below the threshold kz . Indicator kriging estimation is used as probability estimates, without any back transform. Any non-linear back transform of kriging may lead to severe biases if not carefully attended. No robust unbiasedness corrections associated to back transform of kriging estimates may wipe out any benefit brought by working on the transformed variable. In most cases the back transform does not adversely affect the reproduction of the Z-value variogram. Bayes’ theorem relates the conditional and marginal probabilities of events A and B, where B has a non-vanishing probability.

P( B | A )P( A )P( A| B )P( B )

= … … … … … … … … (5.10)

Each term in Bayes’ theorem has a conventional name:

• P(A) is the prior probability or marginal probability of A. It is “prior” in the sense that it does not take into account any information about B

• P( A| B ) is the conditional probability of A given B. It is also called the posterior probability because it is derived from or depends upon the specified value of B

• P( B | A ) is the conditional probability of B given A • P( B ) is the prior or marginal probability of B, and acts as a normalizing constant

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In Indicator kriging when considering the n(u) data altogether as a single multiple-point data event, it identifies Bayes relation.

*SKI ( u ) Pr ob{ I( u ) 1| n( u )}

Pr ob{ I( u ) 1, n( u )}Pr ob{ n( u )}

Pr ob{ I( u ) 1, I( u ) i , 1,... ...,n( u )}Pr ob{ I( u ) i , 1,... ...,n( u )}

α α

α α

αα

≡ ==

=

= = ==

= =

… (5.11)

The probability in the numerator of the above expression is actually a ( n( u ) 1)+ -point, non-centered, covariance, while the denominator is a n(u)-point covariance. 5.4.2. Simple kriging (SK) Simple kriging is used to estimate the porosity of the sand bodies where sand channels are present in the oil reservoir. Accepting a Gaussian model allows building a kriging system to retrieve by kriging the two conditional moments (mean and variance) which suffice to specify the Gaussian conditional distribution. In Simple kriging (SK), the mean of the domain is assumed constant and known. 5.4.3. Ordinary kriging (OK) In Ordinary kriging, the expected value of the random function is locally re-estimated from local data, while the covariance model is kept stationary. The OK concept has been extended to local estimation of the parameters of a functional trend (KT or kriging with a trend) (Goovaerts, 1997). In Ordinary kriging (OK), the mean inside each estimation neighborhood is unknown but constant; it is evaluated by the OK algorithm from the neighborhood data. The difference between the Simple and Ordinary kriging is that SK estimates an unknown deviation from a known stationary mean, say m. If that mean is considered locally variable, it can be estimated from the same local data n(u), then the corresponding estimate will be the Ordinary kriging (OK) (Goldberger, 1962; Matheron, 1970; Goovaerts, 1997). 5.4.4. Non-Linear kriging Most so-called non-linear kriging, including kriging of normal score transform (Deutsch and Journel, 1992) is used in the program SGSIM. The normal score transformation (nst) ranks the dataset from lowest to highest values and matches these ranks to equivalent ranks from a normal distribution. The transformation is defined by taking values from the normal distribution at that rank. The normal score

50

transformation function changes with each particular dataset. Normal score transform can be useful for geostatistics because when the data is dependent, it may be easier to detect and model autocorrelation using the NST. For this reason, the NST must occur after detrending since covariance and semi-variograms are calculated on residuals after trend correction. A nonlinear transform of the original variable(s) is warranted if that transform allows defining new variables that better satisfy the requirements of the algorithm being considered, e.g. the normal score transform allows meeting one requirement of the sequential Gaussian simulation algorithm. The kriging means and variances of normal score values are used directly to build the conditional distributions in the sequential Gaussian simulation, these kriging results are never back transformed; it is the final simulated values that are back transformed, then indeed that back transform is sensitive to the tail extrapolation decision (Deutsch and Journel, 1992; Goovaerts, 1997). 5.5. Sequential Gaussian simulation Sequential Gaussian simulation is a variogram based simulation algorithm. Variogram-based Simulation: Simulated realizations from the algorithms draw their spatial pattern from input variogram model. The variogram-based algorithm is used to simulate reasonably amorphous (high entropy) distributions. Variogram-based sequential simulations have been the most popular stochastic simulation algorithms mostly due to their robustness and ease of conditioning, both to hard and soft data. They do not require rasterizing (Regular or Cartesian) grid, they allow simulation on irregular grids such as point sets. There are several stochastic simulation algorithms, such as

SGSIM (Sequential Gaussian Simulation), SISIM (Sequential Indicator Simulation), COSGSIM (Sequential Gaussian Co-simulation), COSISIM (Sequential Indicator Co-simulation) and DSSIM (Direct Sequential Simulation).

SGSIM, COSGSIM and DSSIM are the choices for most continuous variables, while SISIM and COSISIM are designed for categorical variables. Any set of Gaussian Random Variable (RV) is fully characterized by its mean vector and covariance matrix; all conditional distributions are Gaussian fully characterized by only two moments, the conditional mean and variance given by simple kriging (Journel and Huijbregts, 1978). Thus a Gaussian Random Function (RF) model would appear as the ultimate model when only two-point statistics can be inferred. Unfortunately, the qualifier is that a Gaussian Random Function (RF) maximizes entropy (disorder) beyond the input

51

covariance model; hence a Gaussian-based simulation algorithm such as SGSIM cannot deliver any image with definite patterns or structures involving more than two locations at a time. In the SGSIM algorithm, the mean and variance of the Gaussian distribution at any location along the simulation path is estimated by the kriging estimate and the kriging variance. The value drawn from that distribution is then used as conditioning data. Transform of the original data into a Gaussian distribution may be necessary and is normally performed by the normal score transform. The SGSIM algorithm uses the sequential simulation formalism to simulate a Gaussian random function. Let Y(u) be a multivariate Gaussian random function with zero mean, unit variance and a given variogram model ( h )γ , conditioned to the corresponding Gaussian data (n), can be generated by the following Algorithm 3 (Remy, et al. 2007):

Algorithm 3: Sequential Gaussian Simulation 1. Define a random path visiting each node of the grid 2. for each node u along the path do 3. Get the conditioning data consisting of neighboring original hard data (n) and previously simulated values 4. Estimate the local conditional cumulative distribution function (cdf) as a Gaussian distribution with mean given by a form of kriging and its variance by the kriging variance 5. Draw a value from that Gaussian ccdf and add the simulated value to the data set 6. end for 7. Repeat for another realization A non-Gaussian random function Z(u) must first be transformed into a Gaussian random function Y(u); Z( u ) Y( u )a . If no analytical model is available, a normal score transform should be applied. The simulated Gaussian values should then be back transformed. The Algorithm 3 then becomes Algorithm 4 (Remy, et al. 2007):

Algorithm 4: Sequential Gaussian Simulation with normal score transform 1. Transform the data into normal score space. Z( u ) Y( u )a 2. Perform algorithm 1 3. Back transform the Gaussian simulated field into the data space. Y( u ) Z( u )a

52

Detailed steps in SGSIM are as follows:

Transform data to “normal space” Establish grid network and coordinate system (Zrel -space) Assign data to the nearest grid node Determine a random path through all of the grid nodes

a) find nearby data and previously simulated grid nodes b) construct the conditional distribution by kriging c) draw simulated value from conditional distribution

Check results a) honor data? b) honor histogram: N(0,1) – standard normal with a mean of zero and a

variance of one? c) honor variogram? d) honor concept of geology?

Back Transform [Source: GSLIB]

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Chapter 6 Results and Discussion In this study, one small-size synthetic case and three semi-synthetic cases from the Norne Field is chosen to show an example of automatic history matching. Porosity and permeability are the major parameters for reservoir simulation model which have a significant impact on reserves, production forecasts and economics of the reservoir. In the first step of the flow simulation, forward modeling is used to obtain fluid saturations and pore pressures in the entire reservoir. In this study, a commercial finite difference black oil reservoir simulator (Eclipse 100) (Schlumberger, 2008.1) is used for this purpose where the input reservoir parameters to the simulator are porosities and permeabilities, and output parameters are fluid saturations and pore pressures for each cell at desired time steps. An Ensemble Kalman Filter is used together with the reservoir simulation model for the parameter estimation relating reservoir properties such as pore space, pore fluid, fluid saturation, reservoir pressures, and rock composition. 6.1. Case study The efficiency and accuracy of the presented optimization technique is tested by using different complex, synthetic reservoir models. Two different cases are defined on the basis of geometry and initial ensemble in the present study. Based on geometry two different cases are generated: Case 1 and Norne. Based on initial ensemble four different cases are generated as Case 1 and Norne (Case A , Case B and Case C). Case 1 reservoir model is generated with the properties follow from SPE comparative 3 (Kenyon and Behie, 1987) and Norne model is developed on the basis of a real field data from the Norne field.

54

6.1.1. SPE Comparative 3 It is the paper on Third SPE Comparative Solution Project: Gas Cycling of Retrograde Condensate Reservoirs presented by Douglas E. Kenyon, SPE, Marathon Oil Co. and G. Alda Behie, SPE, Dynamic Reservoir Systems. Original manuscript received in the Society of Petroleum Engineers on October 24, 1983. Paper accepted for publication on October 24, 1986. Revised manuscript received on April 23, 1987. Paper (SPE 12278) first presented at the 1983 SPE Reservoir Simulation Symposium held in San Francisco, November 15-18. 6.1.2. Norne Field

6.1.2.1. General field information The Norne Field is situated in blocks 6508/1 and 6608/10 in the southern part of the Nordland II area in the Norwegian Sea. It is located around 80 kilometers north of the Heidrun Field in the Norwegian Sea (Figure 6.1). The sea depth in the area is 380 meters.

55

Figure 6.1: a) Location of Norne Field in respect with Heidrun Field in Norwegian Sea [Statoil, (2001)] & b) Location of the Norne oil field in the Norwegian Sea (RIGZONE, 2004) In Block 6508/1 (128B), StatoilHydro Petroleum AS is the main operator with participating interests of 39.0% in the Norne Field, whereas Petoro AS has 54.0% and Eni Norge AS has 6.9% respectively. In Block 6608/10 (128), StatoilHydro Petroleum AS has the participating interests of 63.95%, whereas Petoro AS and Eni Norge AS are partners with interests of 24.55% and 11.5% respectively (NPD, 2009). The Norne Field was discovered in the Norwegian Sea in December, 1991. The reservoir is around 3×9 km in extent. It consists of two separate oil compartments, the Norne Main Structure (Norne C-, D and E-Segment) and the Northeast Segment (Norne G-Segment) (Figure 6.2). The Norne Main Structure was discovered in December 1991 and includes 97% of the oil in place.

56

Figure 6.2: The Norne Field segments and wells (Statoil, 2006)

6.1.2.2. General geology of Norne Field The hydrocarbons were found in the rocks of Lower and Middle Jurassic age (Statoil, 2001). The source rocks for the oil and gas in the Norne Field are believed to be the Spekk Formation from Late Jurassic and coal bedded Åre Formation from Early Jurassic (NPD, 2005). The discovery well, 6608/10-2, proved a total hydrocarbon-bearing column of 135 m in the rocks of Lower and Middle Jurassic age. The column consists of a 110 m thick oil leg with an overlying gas cap of approximately 75 m thick which was confirmed by the Well 6608/10-3.

57

Oil is mainly found in the Ile and Tofte Formations. Approximately 80% of the initial oil is located in the Norne Main structure, and the free gas is in the Garn Formation. The reservoir depth is about 2500-2700 meters below the sea surface with a good quality reservoir. The average porosity is in the range of 25-30% while permeability varies from 20 to 2500 mD, net-to-gross values range from 0.7-1 and water saturation 12-43% for the hydrocarbon zones (Statoil, 2001). Figure (6.3) illustrates the lithological description of the reservoir rocks in the Norne Field:

Figure 6.3: Stratigraphical sub-division of the Norne reservoir (Statoil, 2001)

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6.1.2.3. Structure of the Norne Field The Norne Field is a flat horst structure with a trend of NE-SW. The Norne Main Structure is relatively flat with a gas filled in the Garn Formation and the gas oil contact in the vicinity of the Not Formation clay stone. The Not Formation behaves as a cap rock, preventing communication between the Garn and Ile Formations. Acquired reservoir pressure data from the development wells indicate that the Not Formation is sealing and there is no reservoir communication across the Not Formation during production. The northern flank dips towards north-northwest with an oil leg in the Garn Formation (Figure 6.4). The initial gas-oil (GOC) and oil-water (OWC) contacts in the different formations and segments are listed in the Table 1 (Statoil, 1994):

Table 1: Gas-oil and oil-water contacts in the Norne Field C-segment D-segment E-segment G-segment Formation

OWC GOC OWC GOC OWC GOC OWC GOC Garn 2692 2582 2692 2582 2618 2582 2585 No gas cap

Ile 2693 2585 2693 2585 2693 2585 Water filled

Water filled

Tofte 2693 2585 2693 2585 2693 2585 Water filled

Water filled

Tilje 2693 2585 2693 2585 2693 2585 Water filled

Water filled

Figure 6.4: NE-SW running structural cross section through the Norne Field with initial and indications of present fluid contacts, and current drainage strategy (Statoil, 2006)

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As the Norne Field is located on a horst, there are a number of non-sealing faults present which are intra-reservoir faults. It means that the fault displacement is less than the reservoir thickness (Statoil, 2001).

6.1.2.4. Depositional environment The Åre Formation is the lowest formation within the Norne Field and the depositional environment was probably alluvial to delta plain setting, transported from a source area to the east. The Tilje Formation was deposited in a marginal marine, tidally affected environment. The Tofte Formation was in marine condition from foreshore to offshore. The Ile Formation was deposited in the shore face. The Not Formation was deposited in quiet marine environment with influence of the freshwater environment and the Garn Formation was deposited in near shore environments with some tidal influence (Statoil, 1994). 6.1.2.5. Production history of the Norne Filed The amount of initial oil in place is 160.6 mSm3 (Statoil, 2005). The oil production was started from November 6, 1997 by the operator StatoilHydro and gas from 2001. The oil is produced with the drive mechanism of both gas and water injection. Gas injection was ceased in 2005 and all gas is now being exported. However, injection of gas from the C-wells started again in 2006 for an extended period to avoid pressure depletion in the gas cap (NPD, 2008). The total oil reserve is estimated as 510 million barrels. On the basis of NPD’s report on December 31, 2008, the reserve estimates in the Norwegian Share is as follows:

Table 2: Recoverable and remaining reserves in the Norne Field (NPD, 2009) Recoverable reserves:

Oil Gas NGL Condensate mill Sm3 bill Sm3 mill tonn mill Sm3

94.90 11.00 1.70 0.00 Remaining reserves:

Oil Gas NGL Condensate mill Sm3 bill Sm3 mill tonn mill Sm3

14.40 5.20 1.00 0.00 The ultimate oil recovery from the Norne Field until 2016 is estimated to 89.24 million Sm³.

6.1.2.6. Oil enhancement techniques Various measures to increase recovery have been implemented on Norne, including the use of new well technology. A multilateral well was drilled in 2007 which was finished in 2008. The oil production from the Norne Field was at plateau approximately 35 000 Sm3/day, with near full utilization of both oil and gas processing capacity. The field is

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drained with the aid of water and gas injection. But the sub-horizontal shale and calcite permeability barriers and faults have a major impact on reservoir production by gas and water injection. The water injection capacity was increased in 2001 from 42 000 to 55 000 m3/d with the intention of maintaining reservoir pressure and achieving the best possible drainage. A pilot project to evaluate microbial increased oil recovery through the injection of oxygen and nutrient salts (AMIOR) in the reservoir was successful. Other measures for improving oil recovery include processing and interpretation of new 3D seismic with subsequent 4D interpretation for identification of undrained areas in the reservoir and future well location, as well as efficient well solutions for drainage (NPD, 2002). The principle aims of the 4D program over Norne were to accurately map the fluid movements in the reservoir that had resulted from the previous 22 months of production and to provide accurate fluid and rock property information to better understand the reservoir’s past and current behavior. In addition, the information enables forward modeling of the fluid movement so that production plans can be optimized and the life of the field can be extended. 6.2. Synthetic and semi-synthetic case 6.2.1. Case 1 The first case, Case 1, is a 2D gas condensate reservoir with three phases. It is a small-sized synthetic reservoir model with 300 cells (20×1×15). All of them are active. The blocks have equal 250 meters sides in both x- and y-directions. The heights of the grid blocks are varying. Two water injector and one producer wells are situated at the two corners and in the middle part of the reservoir respectively. The properties of the wells are inherited from the producer well at SPE 3. Tables 3 and 4 give the specification of the reservoir model and well input data for Case 1 and Figure (6.5) shows the reservoir model grid and well perforations:

Table 3: Reservoir Grid and Saturation Input Data for Case 1 Reservoir Grid Data: DX = 20, DY = 1, DZ = 15 DX = DY = 250 meters Number of total cells 300 Number of active cells 300 Datum depth, meters 2070 Porosity (at initial reservoir pressure) 0.2 Permeability at initial condition, mD 1000 Datum Pressure, Bar 270 Oil-water contact, meter 2094 Gas-oil contact, meter 2024 Capillary pressure at contact, for water-oil & oil-gas, psi

0

Initial pressure at contact, Bar 300.6

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Water properties: density of oil at surface conditions, kg/m3 800 density of water at surface conditions, kg/m3 1022 density of gas at surface conditions, kg/m3 0.9907 compressibility, 1/Bar 0.000053 PV compressibility, 1/Bar 0.000041

Table 4: Well Input Data for Case 1 Production Well Data: 1 production well I = J = 10 Perforations K = 5, 8 (bottom layers) Well bore diameter, meter 0.5 Bottom hole pressure, Barsa 400 Injection Well Data: 2 injection wells I = J = 1 and I = J = 20 Perforations K = 1, 15 and K = 1, 15 Well bore diameter, meter 0.5 Bottom hole pressure, Barsa 400

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Figure 6.5: a) Reservoir model grid and b) Well perforations for Case 1

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The reference porosity and permeability for this reservoir are 15-31% and 500-1600 mD respectively. Water is mainly injected to accelerate the production. Figure (6.6) shows the real porosity and permeability for Case 1 reservoir model.

Figure 6.6: a) Reference or real porosity and b) Real permeability for Case 1 The production schedule is divided into two sections. The first section of production lasts 2.5 years (from 6 January, 1996 to 17 October, 1998). During this period, the wells are controlled using a target oil rate of 10000 Sm3/day. If the bottom hole pressure goes below 50 bar, this value will be used as target pressure. The second section consists of 7 years of production started from 16 March, 1999 and ended on 8 January, 2006. During this period, the wells are operated on different oil rates starting from 7157.059 and end at 0.472759 Sm3/day. All are considered as a history matching phase. The Ensemble Kalman Filter was also running at the same time in a history matching mode, which means that the wells are steered according to the observed values generated by running the simulator using the “true” permeability and porosity fields. Measurements from all the three wells were used during the assimilation period. This includes bottom hole pressures, gas/oil ratios and water cuts. As a reservoir parameter, water saturation and pore pressures in each cell are considered independent. It is guessed that there is no depletion (fluid flow changes) or no pore pressure changes in the reservoir at the initial stage. Figures (6.7) & (6.8) show the values of reservoir pore pressure and saturation in each cell. In the beginning, the pore pressure ranges from 266 bar to 277 bar, but at the end of the production, in the middle part of the reservoir, the pressure decreases to 235 bar and in the corner the pressure increases from 340 bar to 400 bar.

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Figure 6.7: Initial guess of reservoir pore pressure a) Beginning and b) End of production

Figure 6.8: Initial guess of reservoir water saturation a) Beginning and b) End of production 6.2.2. Norne The second case is a 2D semi-synthetic model which is a cross section of a real field in the North Sea called the Norne Field. The total reservoir model contains 99372 (39×98×26) cells where the cross section is generated from 50th cell at y direction (Figure 6.9).

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Figure 6.9: Cross-section of the Norne Field in 50th cell Norne is a two-dimensional small-size semi-synthetic reservoir model developed on the basis of a real field data from the Norne Field. The model has corner point geometry and contains 1014 grid blocks (39×1×26) of which 739 are active. This model has one producer well and one water injector well that are located at columns 6 and 36 respectively. The properties of these wells are inherited from the well F-4H that is situated at block number 49 in the y direction in the full field model. Table 5 shows the specification of the reservoir model and fault data and Table 6 presents well input data for Norne semi-synthetic case. Figure (6.10) represents the reservoir grid, major faults and well perforations of Norne Field.

Table 5: Reservoir Grid and Fault Data Reservoir Grid Data: DX = 39, DY = 1, DZ = 26 Number of total cells 1014 Number of active cells 739

Datum depth

(meters)

Datum Pressure (Bar)

Water-Oil contact (m)

Gas-Oil contact (m)

2582 2500 2582 2200 2585

268.56 263.41 269.46 236.92 268.77

2692.0 2585.5 2618.0 2400.0 2693.3

2582 2500 2582 2200 2585

Reference pressure, Bar 277 density of oil at surface conditions, kg/m3 860 density of water at surface conditions, kg/m3 1033 density of gas at surface conditions, kg/m3 0.853 Reference water formation volume factor, rm3/sm3

1.038

Rock compressibility, 1/Bar 4.84E-5 Water compressibility, 1/Bar 4.67E-5 Reference viscosity of the water, cP 0.318

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Fault Data: Number of Faults: 5 Name IX1 IX2 IY1 IY2 IZ1 IZ2 Face

DE 14 14 1 1 1 26 ‘X’ DI 20 20 1 1 1 26 ‘X’ GH 29 29 1 1 1 26 ‘X-’

E3_3 9 9 1 1 1 26 ‘X’ G 33 33 1 1 1 26 ‘X’

Fault segment 35 Table 6: Well Input Data for Norne Production Well Data: 1 production well

Well I & J K1 K2 P1 6 1 -14 1 -14

Injection Well Data: 1 injection wells I1 36 5 -26 5 -26

Well bore diameter, meter 0.216 Bottom hole pressure, Barsa 300

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Figure 6.10: a) Reservoir grid and major faults and b) well perforations of Norne Field

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The reference porosity in this reservoir is in the range of 20-30% while permeability varies from 20 to 2000 mD (Figure 6.11).

Figure 6.11: a) Real porosity and b) Real permeability for Norne reservoir model The injection well is controlled by reservoir fluid volume rate. Water is mainly injected to accelerate the production. The bottom hole pressure is fixed at 300 bar. The production schedule is divided into two sections. The first section lasts 20 years and is considered as a history matching phase. The history matching period is started from 11 November, 1987 and ends at 01 November, 2007. During this period, the production well has the production rate ranges from 50.22773 to 223.1928 Sm3/day. The Ensemble Kalman Filter was also running at the same time in a history matching mode, which means that the wells are steered according to the observed values generated by running the simulator using the “true” permeability and porosity fields. The second section consists of 10 years of production forecast in which the porosity and permeability of each ensemble has been considered to run the simulation. In this case, from 100 ensemble data, the porosity and permeability have taken into account and that gives the forecast for another 10 years starting from 1 November, 2007 to 1 November, 2017. In the Norne Field, water saturation and pore pressures in each cell are also considered independent. Figures (6.12) and (6.13) show the values of reservoir pore pressure and saturation in each cell. In the beginning, the pore pressure ranges from 260 bar to 290 bar, but at the end of the production, the pressure ranges from 257 to 306 bar where the pressure is maintained by injecting water.

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Figure 6.12: Initial guess of reservoir pore pressure a) Beginning and b) End of production

Figure 6.13: Initial guess of reservoir water saturation a) Beginning and b) End of production 6.3. Ensemble Kalman Filter for this study 6.3.1. Introduction The Ensemble Kalman Filter (EnKF) is an estimation method, which performs continuous statistical updating of parameters and dynamical states, and estimates the model uncertainty by using an ensemble of model representations. The Matlab code is developed in the International Research Institute of Stavanger (IRIS) on unix/linux platforms with Eclipse as the reservoir simulator to run EnKF (Flornes, et al. 2007).

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6.3.2. Input files The EnKF input files are all the files required to run the Ensemble Kalman Filter Matlab code. The input files are binary “MAT-files” named “filename.mat”. There are two binary Matlab files that have to be specified before running the automated EnKF code. These are named inputData.mat and trueSolution.mat. Each of these files contain a number of individual variables. Some of these variables are structures that can contain several fields. In addition to these two input files, it is necessary to specify the initial ensemble of the variables/parameters which will be updated/estimated by EnKF. The initial ensemble is a matrix with a number of rows equal to the number of ensemble members that can be generated using a statistical package and saved in a .mat file. The initial ensemble is constructed based on prior geostatistical information. For Case 1, as it is a synthetic case, the initial ensemble is prepared followed by the SPE comparative 3 paper. Case A and Case B of the Norne Filed are considered randomly where Case C has been constructed depending on the basis of SGSIM. A semi-automatic script called setupCase.m is used which is semi-automatic in the sense that it has to be manually edited to fit the example at hand, and then it can automatically generate inputData.mat and trueSolution.mat. The main steps in the script setupCase.m are:

1. To make inputs to be saved in inputData.m (a) Set the options, (b) Set the kalmanOptions, (c) Set numIter, (d) Save inputData.mat.

Regarding 1(a) and 1(b): The options are set by specifying options as an empty variable and then filled it by scanning from the Eclipse input file.

2. Make inputs to be saved in trueSolution.mat (a) Perform an Eclipse forward run through all the time steps, (b) Save results from this forward run in a .mat file, (c) Use the results from the forward run to specify the H matrix and the W matrix, (d) Create the cell array measurements, (e) Save trueSolution.mat.

Regarding 2(a) to 2(e): For a synthetic case, a forward run of Eclipse through all the time steps must be performed to compute synthetic well measurements. For field cases, the measured well data are usually contained in the “SCHEDULE” section of the Eclipse input file, the well data are listed after each “DATES”/”TSTEP” specification. So, for a real field example, the measurements have to be scanned from the Eclipse input file and saved correctly in the variable named measurements. For a real field case, it is not absolutely necessary to run a

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forward run before performing the EnKF updating, but it is good to do that. A forward run of Eclipse is performed to generate the Eclipse output file. H and W are manually specified in the same way as setupCase.m is made. The forward run will make it easier to specify H and W.

3. Edit and run setupCase.m (a) Manually edit the script setupCase.m up to the line where inputData.mat are being saved, (b) Run setupCase.m until the forwSim.mat file is saved, (c) Load the matrix y from forwSim.mat and use this matrix when manually specifying the matrixes H and W in setupCase.m, (d) Run setupCase.m again, now both inputData.mat and trueSolution.mat should be correctly generated.

The list of the static variables that are updated is porosity and horizontal permeability (‘PORO’, ‘PERMX’) which are constant when Eclipse runs forward and are updated at each time step by EnKF. The state or dynamic variables are pressure, saturation of water and gas and the gas-oil ratio ('PRESSURE', 'SWAT', 'SGAS', 'RS'). The state / dynamic variables and well measurements are also updated by EnKF. The initial ensemble for the dynamic variables do not specify at the first time step (beginning of the reservoir life production) as there are no dynamic variables and the fluids are initialized to potential equilibrium. The H matrix is a cell array with a number of matrices equal to the length per time-spec, i.e., one matrix for each time spec. Each matrix has two columns and contains indices specifying which of the rows of the matrix called simulatedEnsemble that should be used as measurements in the EnKF updating or it is a matrix containing indexes specifying which of the rows of the predictedState matrix that should be used as measurements in the history matching. W is also a cell array and is a measurement covariance matrix for the specific observation time. Hones=1 allows to specify the H matrix and the covariance matrix, W, manually. When Hones=1, computeTrueSolution is run twice, first to run Eclipse, then to add noise to the measured data. For a synthetic case, there must have a forward simulation that represents the “true” case. “True” measurements are considered for the synthetic and semi-synthetic case. In this measurement, the data is considered in similar to the real field data and make the synthetic case in such way that will more or less represent the real condition of a field. This is the purpose of forwSim.mat. y is a variable that has one column for each timestep. In Eclipse data file, in the SUMMARY section, the variables are specified in rows. For specifying H, this matrix y is not needed, but it might be needed to specify W. If the uncertainty of the measurements are relative (given in percent), the value of y needs to be multiplied with the relative value to get an absolute uncertainty for specifying W.

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6.3.3. Output files The output files are the files automatically generated when running the IRIS - EnKF Matlab code. The output files are, as the input files, binary “MAT-files”. The output files generated during running the EnKF method are named simEns#.mat, result#.mat, and ensemble#.mat. Description of these are given in the EnKF manual (Flornes, et al. 2007). When running the EnKF method on a reservoir example, the following work flow is performed:

(1) Forward step: Eclipse is run from time step t to time step t + 1 (2) Analysis step: EnKF is used to update the free parameters, the dynamic variables, and the static variables calculated by Eclipse (3) Use result from (2) as input to Eclipse, set t = t + 1, and return to 1

For each time (1) to (3) is performed, the three result files are saved. Hence, three files are saved for each time step that is indicated by a number appearing in the result files illustrated by #, starting from 0. To be able to give the dimensions of the variables contained in the result files, the following numbers are needed:

K = (f + s · options.fieldSize + d · options.numGridBlocks + k)… … … (6.1) where,

• f is the number of free parameters to be estimated • s is the number of static variables • d is the number of dynamic variables • options.fieldSize is the total number of grid blocks in the reservoir • options.numGridBlocks is the number of active grid blocks in the reservoir • k is the number of well data calculated by Eclipse

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The following are the flow charts which show the flow of the work done by the Ensemble Kalman Filter :

Figure 6.14: IRIS - EnKF Matlab work flow description (Flornes et al. 2007)

Matlab Code

generate inputs

setupCase.m

run the main script

runEnKF.m

Evaluate results

Plot figures and Statistics files

return

result files

ensembleXX.mat resultXX.mat simEnsXX.mat

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Figure 6.15: Preparing the Matlab input files. Work flow of setupCase.m (Flornes et al. 2007)

no

no

yes

Inputs

manually edit setupCase.m

to fit the example before executing

setupCase.m

setupCase.m OK

inputData.mat generated

computeTrue Solution.m

return

exist trueSolution.mat

yes

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Figure 6.16: Description of the main script (Flornes et al. 2007)

no

runParSim.m

all time steps

done

return

yes

runEnKF.m

compute Measurement.m

yes

Parallel simulation

runForward Simulations.m

EnKF.m

no

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Figure 6.17: Compute reference solution (Flornes et al. 2007) 6.4. Discussion of the cases on the basis of EnKF 6.4.1. Case 1 The following measurements are used in the EnKF updating for Case 1: Measurements of production rates for oil, water and gas (OPR,WPR and GPR), together with bottom hole pressures (BHP) are collected. The measurements are generated by running the simulator with the true porosity and permeability. 100 initial ensembles for this case are generated depending on the unconditional Gaussian simulation. In this realization the static variables in the initial ensemble are defined with a mean equal to the true static variables and a standard deviation of 0.1 for porosity and 0.5 for the log permeability. The initial ensemble should then span the solution. The correlation length was initialized with a mean of [5, 5] cells in the [x, z] direction that is used to generate initial ensemble for the static variables and standard deviation of 1 cell. Standard deviation

Reference Solution

computePredictedSolution.m

Use input arguments

-1 -1

load inputData.mat

load result0.mat

forwardeclipse

return

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(SD) in correlation length used to generate initial ensemble for the static variables. This will initially give a scattered ensemble but spanned around the true solution (Figure 6.6). In Ensemble Kalman Filter, the uncertainty is calculated directly by the standard deviation of the ensemble and the measurements. For getting the output values, the setup case for the ensemble Kalman filter is set in the following way: numIter is 27 which is a scalar value that gives the number of time steps the ensemble is going to run, trueStaticVar is 600 which is a vector that contains true values of the static variables. The static variables are usually porosities and permeabilities which are written as a set of files, one for each of the “n” variables to be updated (i.e., porosity and/or permeability in the x, y or z-directions are treated as different variables). trueSolution contains information regarding the measurements to be used in the updating/estimation. In this file, matrix with numIter columns that correspond to a certain observation time and lists the static variables, the state variables and the measurements respectively is given 1844×27. After running the setupCase.m file in Matlab, it gives well measurements during the EnKF updating (Figure 6.21a). Then predictedSolution can be computed from a given time step to another given time step. This function is used to compute the reference solution if the input parameters are set to “-1,-1” that gives the plot of true static fields and reference fields (Figure 6.18). The reference solution is a forward simulation of the mean of the initial ensemble without any filter applied. In the figure, the true porosity and true permeability in the x-direction (PERMX) of this synthetic example is shown where the permeability in the y-direction (PERMY) is equal to the permeability of the x-direction and that the permeability in the z-direction (PERMZ) is equal to 0.1 times the permeability in the x-direction.

Figure 6.18: True static and reference field of porosity and permeability from beginning and end of EnKF updating where,

• Color axis of the permeability is given in mD • Ref sol represents mean of initial ensemble • True poro and perm represents the true field for true porosity and permeability

In Figure (6.18), the reference porosity shows 20% of porosity in the beginning and approximately ±20% at the end and the reference permeability shows 1000 mD which is also the same as the true permeability as the true porosity and true permeability in the initial ensemble was set to 20% and 1000 mD respectively.

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Forecasts with initial ensemble members are computed where numStep must be set to -1 to use initial ensemble members and to choose 100 first members of the ensemble to run the forecasts to get reference solution and forecasts with initial ensemble members (Figure 6.21b). After that ensemble Kalman filter can be run with Eclipse. To plot estimated static fields and standard deviation in estimate, for example plotEstStaticVar (5,[10 20 30 40 49]) is used where the input argument decides how many plots of each estimated static field along the time axis are plotted. First all the estimated fields are plotted, then the standard deviation of the fields are plotted in the same order. In Case1, the numStep is equal the time step which is 27. Here if 5 indicates the number of plots of each estimated static field along the time axis, then in Case 1 it would be (3,[10 20 26]) which gives plots with respect to true porosity, then permeability and then the standard deviation of the fields (Figure 6.19 - 6.20).

Figure 6.19: Estimated static fields for porosity and standard deviation for Case 1

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Figure (6.19) shows estimated static fields for porosity and standard deviation from the beginning and end with respect to reference solution in different time (days) after running the ensemble Kalman filter. The forecast shows a small increase of porosity in days 2400 in the beginning that ends up with more than 30% porosity at the end of EnKF updating with respect to the reference solution. The associated result of the standard deviation shows reduction (±0.06 to ±0.08) in the edge, whereas it gives comparatively lower value in the middle where the production well is situated. In true condition SD is set to 0.1. Figure (6.20) represents Estimated static fields for permeability and standard deviation from beginning and end with respect to reference solution in different time (days) during EnKF updating. The forecast represents 700 mD – 1000 mD of permeability from days 2400 in the beginning that ends with 1000 mD in the edge compared with true measurements. The associated standard deviation in the beginning shows comparatively lower value (±0.2 to ±0.3) from days 2400 in the beginning and increased at the end of the EnKF update where the SD is set to 0.5 for log permeability.

Figure 6.20: Estimated static fields for permeability and standard deviation for Case 1

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Then forecasts with estimated ensemble members are computed when the numStep is equal to the time step of the last time step to run for the forecasts. For this case the last time step is 26. It will give the reference solution and forecasts with estimated ensemble members (Figure 6.21c).

Figure 6.21: a) True well measurements used in the Ensemble Kalman updating, b) Reference solution and forecasts with initial members for the wells and c) Reference solution and forecasts with estimated ensemble members for Case 1

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In this figure, • Black solid line represents synthetic true measurements • Red dots represent measurements with added noise • Green line represents the result of a forward run of Eclipse by using mean of initial

ensemble (ref sol) as the static fields • Blue line represents forward run of Eclipse using the initial ensemble members as

input for static fields

For Figure (6.21c), • Blue line represents forward run of Eclipse using the ensemble members updated

at time step “numStep” as input for static fields • Orange line produced by using the mean of the updated ensemble for the static

fields at each time step From Figure (6.21), it can be seen that the Blue line that represents forward run of Eclipse using the ensemble members updated at time step as input for static fields (Figure 6.21c) gives a better match than the same Blue line represents forward run of Eclipse using the initial ensemble members as input for static fields (Figure 6.21b). The Orange line which is the mean of the updated ensemble for the static fields at each time step clearly match the reference solution of Black solid line that represents synthetic true measurements (Figure 6.21a & c). 6.4.2. Norne The following measurements, collected from the open wells, are used in the EnKF updating for Norne Field: Measurements of production rates for oil, water and gas (OPR, WPR and GPR), together with bottom hole pressures (BHP). The measurements are generated by running the simulator with the true permeability. For getting the output values, the setup case for the ensemble Kalman filter is set in the following way: numIter is 39 for running the ensemble, trueStaticVar is 2028 contain true values of the static variables of porosity and permeability. In this file, in trueSolution, matrix with numIter columns that correspond to a certain observation time and lists the static variables, state variables and the measurements respectively is given 5043×39. The Norne case is divided into three different cases based on the initial ensembles.

6.4.2.1. Case A Norne Case A is using the same amount and the same initial ensembles which are used in Case 1 of the synthetic model. 100 initial ensembles for this case are generated depending on the unconditional Gaussian simulation. In this realization the static variables in the

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initial ensemble are defined with a mean equal to the true static variables and a standard deviation of 0.1 for porosity and 0.5 for the log permeability. The initial ensemble should then span the solution. The correlation length was initialized with a mean of [10, 10] cells in the [x, z] direction that is used to generate initial ensemble for the static variables and standard deviation of 0 cell. Standard deviation in correlation length used to generate initial ensemble for the static variables. This will initially give a scattered ensemble but spanned around the true solution (Figure 6.11).

6.4.2.2. Case B Norne Case B like Case1 and Norne Case A also uses unconditional Gaussian simulation to generate 100 initial ensembles for this case. In this realization the static variables in the initial ensemble is defined with a mean equal to that of the true static parameters and a standard deviation of 0.1 for porosity and 0.5 for the log permeability. The correlation length was initialized with a mean of [10, 10] cells in the [x, z] direction that is used to generate initial ensemble for the static variables and standard deviation of 0 cell.

6.4.2.3. Case C In this realization the static variables in the initial ensemble are defined by the RMS (Risk Management Solution) tools where the mean and standard deviation is based on the measurements made in the wells. Case C is using the same amount but completely different kind of initial ensembles. The Stanford Geostatistical Modeling Software (SGeMS) is used to generate one thousand multiple realizations of the porosity fields which are conditioned to the available well data and variograms. Details of SGSIM can be found in Chapter five. The same amount of the horizontal permeability realizations are generated based on geological relationship between permeabilities and porosities in different geological sections. Vertical permeabilities are set to be equal with horizontal permeabilities. Figure (6.22) represents some of these ensembles.

(a) (b)

Figure 6.22: Some of multiple realizations of (a) porosity and (b) permeability (Dadashpour et al. 2009)

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Considering the same way of running Case 1, all cases of Norne have been run to compare the outputs which have shown below in Figure (6.23 to 6.28).

Figure 6.23: Comparison of the true static and reference field of porosity and permeability for Norne In Figure (6.23), comparison of the true static and reference field of porosity and permeability from beginning and end of EnKF updating for Norne field is given where the true porosity for all three cases of Norne rages from 24% - 32% whereas true permeability ranges from 250mD - >1000 mD.

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Figure 6.24: Comparison of the estimated static fields for porosity and standard deviation for Norne In Figure (6.24), comparison of the estimated static fields for porosity and standard deviation from beginning and end with respect to reference solution in different time (days) for Norne is shown. Here the comparison of the forecast shows that the beginning and end porosity for Case A is 30% - >40% with standard deviation of 0 - 0.08. The porosity in Case B ranges from 20% – 45% with standard deviation of 0 – 0.08. For Case C, the porosity ranges from 22% - 28% and standard deviation of 0 – 0.02. The true porosity for the three cases are 30% for Case A, 25% - 30% for Case B and 24% - 32% for Case C respectively gives a better match with the forecast. Here Case C has the minimum standard deviation compare to the other two cases.

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Figure 6.25: Comparison of the estimated static fields for permeability and associated standard deviation (SD) from for Norne In Figure (6.25), the comparison of the estimated static fields for permeability and associated standard deviation (SD) from beginning and end with respect to reference solution in different time for Norne is given. The development in the estimated permeability is shown together with the true permeability and reference solution. The reference solution is the mean of the initial ensemble. The comparison shows that Case A has the permeability of 250 mD – 1000 mD with SD 0.2 - 0.4; whereas Case B and Case C have the permeability of 250 mD – 1000 mD and 250 mD - 2000 mD respectively with the standard deviation of 0.3 - 0.4 for Case B and 0 - 0.6 for Case C respectively. In Case A and Case B, the end estimated permeability mismatch with the true permeability whereas in Case C, one can observe that after 720 days, a good match of the permeability field is obtained in reference with the true condition with a higher permeability of 2000 mD.

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Figure 6.26: True well measurements used in the Ensemble Kalman updating for Norne

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In Figure (6.26): • Black solid line represents synthetic true measurements • Red dots represent measurements with added noise

Comparison among the three cases of Norne shows true measurements. In Figure (6.27):

• Red dots represent measurements with added noise • Green line represents the result of a forward run of Eclipse by using mean of initial

ensemble (ref sol) as the static fields • Blue line represents forward run of Eclipse using the ensemble members updated

at time step “numStep” as input for static fields Comparison shows that Case C has the higher water production rate than the mean of the initial ensemble (ref sol) from the beginning of the production, but Cases A and B show higher values of water production after 3653 days. In Figure (6.28):

• Red dots represent measurements with added noise • Green line represents the result of a forward run of Eclipse by using mean of initial

ensemble (ref sol) as the static fields • Blue line represents forward run of Eclipse using the initial ensemble members as

input for static fields • Orange line produced by using the mean of the updated ensemble for the static

fields at each time step

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Figure 6.27: Reference solution and forecasts with initial members for three cases of Norne

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Figure 6.28: Reference solution and forecasts with estimated ensemble members for Norne

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From the above three figures (Figures 6.26 to 6.28), it can be summarized that Figure (6.28) gives better match with the true condition (Figure 6.26). It can also be seen from the above figure that, Case C gives better results compared to Case A and Case B. Case A and Case B gives a better match with the true measurements for production of oil, water and gas with increased bottom hole pressure during production. On the contrary, Case C gives small reduction in the production of oil and gas with increased production of water in 3563 days where the bottom hole pressure for both the production and injection well remain the same with the true condition. For parameter estimation, comparison of the real and mean porosity and permeability for the synthetic Case 1 shows (Figure 6.29 and 6.30) a difference for both the porosity and permeability with the real case.

Figure 6.29: Comparison of the real and mean porosity for Case 1

Figure 6.30: Comparison of the real and mean permeability for Case 1 For Norne, comparisons were made for all the three cases compared with the real case (Figure 6.31 & 6.32) which shows that for both the porosity and permeability, Case C has very close similarity compared to Case A and Case B with the real case.

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Figure 6.31: Comparison of the real and mean porosity for Norne

Figure 6.32: Comparison of the real and mean permeability for Norne

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For predicting future performance of the reservoir from the Ensemble Kalman Filter, cumulative oil production of the initial and true ensemble members are considered together with the other ensembles. After running the ensemble Kalman Filter with Eclipse, the results of 100 ensembles where each different initial ensemble is compared in order to investigate the robustness. The ensemble size plays an important role in determining the degree of stability of the ensemble Kalman filter. More the ensemble size gives better results. Here observed data are assimilated for 10 years and predict until next 10 years to see the effect of data assimilation in the late production history. For getting these measurements, updated EnKF data for porosity and permeability have been considered for each ensemble members to run eclipse and forecast for getting the observed and prediction values. These values have been turned to cumulative values to produce the graphs that are shown below in Figure (6.33 – 6.39). In all the figures, production data generated from the ensemble are shown as light blue curves; data generated as true values are shown in red dots and pink boxes for the production and future value respectively and initial data and forecasts are shown in purple color solid and broken line respectively.

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Figure 6.33: Cumulative oil production where acceptable simulations (broken lines) were used to forecast future productions of next 10 years for Case 1 In Figure (6.33), the oil production rate in the beginning goes to 80 000 Sm3/day and it has gone to ~ 130 000 to 150 000 Sm3/day within 3.5 years of production. The rate becomes stabilized for the next 20 years where 10 years from the beginning data show the original production and next 10 years show the forecast. Here the true and the initial data show the similar values with the rate of ~ 140 000 Sm3/day. It can be shown from the figure that, after assimilation of production data, the band of predictions increases compared to the predictions from the true and initial ensembles. That means most of the realization members over predict the cumulative oil production during the forecasting period whereas

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the cumulative bottom hole pressure and water cut is similar to the true values (Figure 6.34 & 6.35).

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Figures (6.36 to 6.38) show the cumulative oil production rate for the three cases of Norne. In the following figures first 20 years show the production and next 10 years shows the forecast. In Figure (6.36), the oil production rate started rising from the very beginning of the production year from 2 000 to 12 000 Sm3/day, where most of the ensembles show the range of ~ 6 000 to ~ 9 000 Sm3/day. The true and initial data show similar values to the rate of ~ 7 000 Sm3/day. Some of the ensembles show the rise of the rate from 3 years after production with a rate of ~ 12 000 Sm3/day. In Figure (6.37), the oil production rate started rising in a range from ~ 6 400 to ~ 10 000 Sm3/day, where most of the ensembles show the range of ~ 6 400 to ~ 8 000 Sm3/day. The true and initial data show the similar values with the rate of ~ 6 400 to ~ 6 800 Sm3/day. One of the ensemble shows the rise in the oil production with a rate of ~ 18 000 to 20 000 Sm3/day. In Figure (6.38), the oil production rate has the range from ~ 6 000 to ~ 9 000 Sm3/day, where most of the ensembles show the close ranges with each other. The initial value is a bit lower than the real or true value after 13 years of the production. From the above three figures (Figures 6.36 to 6.38), it can be concluded that for the three cases, the production rates are almost similar compare with the individual ensemble behavior with a rising tendency during the forecasting period. But most of the ensembles from Case B and Case C have very close values compared with the true or real values. Here Case C shows very close values compare with the real value for most of the ensembles during the observed period, but the forecast overpredicts the production rate which is comparatively lesser than the other two cases.

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Figure 6.36: Cumulative oil production for Norne Case A where acceptable simulations (broken lines) were used to forecast future productions of next 10 years

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Figure 6.37: Cumulative oil production with forecast for Norne Case B

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Figure 6.38: Cumulative oil production with forecast for Norne Case C The following figure (Figure 6.39) represents the comparative study of the bottom hole pressure for the injection and production well. Here in all cases, the initial bottom hole pressure for injection well is lower than the true value and other ensemble values, where the production well shows the same initial and true values. For all cases the pressure is maintained to get better recovery by injecting water.

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Figure 6.39: Comparative figures show cumulative bottom hole pressure with forecast for injection and production well in all the three cases of Norne

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The measurement of the mean cumulative values for the production is cheaper and less time consuming in the sense that it is not necessary to calculate forecasts for all ensemble members. Figure (6.40 to 6.44) show the forecast resulting from the cumulative mean of the ensemble compared with the true and initial cumulative values of oil, water and gas production together with bottom hole pressure and water cut for Case 1. The mean cumulative oil production rate (Figure 6.40) shows slightly higher values (~ 147 000 Sm3/day) than the true and initial values (~ 140 000 Sm3/day), whereas mean water production (Figure 6.41) shows the rise of the production at the end of 20 years with a rate of ~ 3 100 000 Sm3/day that is similar to the true and initial values. Mean gas production [Figure (6.42)] rate shows ~ 15 600 000 Sm3/day compare with the true and initial value of ~ 16 500 000 and ~ 15 300 000 Sm3/day respectively. From the first 2 years of production, in case of mean bottom hole pressure, true and initial values remain the same where the initial values getting higher with ~ 7 000 Bar compare to the mean and true BHP values (~ 5 000 Bar) (Figure 6.43). It can be said that the mean production rate is very close sometimes match (for water production) the true values.

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Figure 6.40: Cumulative mean oil production with forecast compare to true and initial data for Case 1

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Figure 6.41: Cumulative mean water production with forecast compare to true and initial data for Case 1

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Figure 6.42: Cumulative mean gas production with forecast compare to true and initial data for Case 1

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Figure 6.44: Mean cumulative water cut for Case 1

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Figures (6.45 to 6.48) represent the forecast resulting from the cumulative mean of the ensemble compare with the true and initial cumulative values of oil and water production and bottom hole pressure for both injection and production wells of the Norne field. The mean cumulative oil production and forecast shows similar values (~ 6 500 to 6 800 Sm3/day) for all the three cases of Norne together with the true and initial values (Figure 6.45). Water production rate (Figure 6.46) are same (~ 2 200 Sm3/day in 20 years time) for all the values during the production time, but the forecast shows that in 21 years, Case B has the higher water production rate (~ 2 400 to 3 600 Sm3/day) than the other two cases which ended up with ~ 5 600 Sm3/day. After 10 years of forecast in water production, Case A and Case B give the production rate of ~ 4 700 Sm3/day. After 5 years of production, the initial value has higher production rate compared with the values of true and all three cases and the forecast shows the rate of ~ 5 100 Sm3/day. The cumulative bottom hole pressure also shows the rise of pressure in the forecast for Case B compare to the other cases (Figure 6.47). All the following figures (Figure 6.45 to 6.48) indicate that Case C has a very good match with the true values of the production and with the pressure which indicates very low uncertainty.

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Figure 6.45: Cumulative mean oil production with forecast for Cases A, B and C for Norne comparing with the true and initial values

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Figure 6.46: Cumulative mean water production with forecast for Norne

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Figure 6.47: Mean cumulative bottom hole pressure with forecast of the injection well for Norne

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Mean Cumulative Bottom hole Pressure with Forecast of Production Well for Norne

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Figure 6.48: Mean cumulative bottom hole pressure with forecast of the production well for Norne Forecast of a single ensemble member of the lowest error value (lowest objective function) is chosen to compare with the mean values in production, pressure and water cut since the mean of the ensemble could possibly contain features that are not present for individual ensemble members. Also in the above figures for both cases (Case 1 and Norne), true cumulative oil production has given as a reference. All forecasts are evaluated at 7312 days (20 years) for Case 1 and at 10900 days (29.8 years) for Norne. For Case 1, comparison between calculated and observed data of 100 ensembles have been considered, where ensemble number 78 shows the lowest value that gives the best iteration compared with the true and initial guess. Considering the porosity and permeability from all ensemble, Eclipse has given RSM and forecast values for producing the graph of WOPR, WBHP and WWCT. Also from all simulated ensembles, the mean of the porosity and permeability has been considered to produce the results of WOPR, WBHP and WWCT (Figure 6.49).

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Figure 6.49: Comparison of calculated lowest objective function and mean values together with forecast in oil production, well bottom hole pressure and water cut for Case 1

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From Figure (6.49), comparisons of the two graphs show similar patterns and values with the true and initial value except the bottom hole pressure where the initial value is much higher than the ensemble values. For Norne, comparison between calculated and observed data gives the best iteration in ensemble number 88, 36 and 7 for Cases A, Case B and Case C respectively compared with the true and initial values. Therefore, similar graphs (Figure 6.50) including forecast of lowest objective function and mean values for oil and water production and bottom hole pressure in injection and production wells have been made. All three cases show similar values with the true measurements. Oil production shows higher values than the initial value in both the graphs but match with the true measurements. Among the three cases, Case C gives the best fit with the true condition in case of oil and water production.

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Figure 6.50: Comparison of lowest objective function and mean values with forecast in oil production, water production and well bottom hole pressure of injection and production wells for Norne The objective function is a function associated with an optimization problem where the object is to find the best of all possible solutions. It measures the misfitting of two things when they are shifted relative to one another. In this study the results of the ensemble Kalman filter is considered where Case 1 shows 26 time steps and Norne shows 37 time steps. From the results of the EnKF for every case, the porosity and permeability are considered to run Eclipse to get the real error of each time step which is put in an Excel file to construct the graph of objective function versus time step.

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Figure 6.51: Objective function with the number of time steps for Case 1 The above figure (Figure 6.51) shows the real error measurements at each time step for Case 1. The graph decreases abruptly in the beginning from 10% to 5% and then it again rises to 6.5% and started decreasing nearly 4%. It decreases until becoming stationary (up to small fluctuations) at a small value after 26 time steps. Figure (6.52) indicates the comparison of the three cases for the Norne where Cases A and B have 10% error compared to Case C of 17% in the beginning of time steps. Case A has 11.5% of error during 27 time steps while Case B and Case C show 3% and 0.5% error respectively. All the three cases becoming nearly zero after 38 time steps. Among the three cases, Case C shows less error which gives better results compared with Case A and Case B.

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Table 7 : Standard Deviation and Error Estimation for Case 1 Initial After Optimization Name SD Error SD Error

Porosity (φ) 0.067452 10.06711 7.09273 10.67757 Permeability (k) 41.0305 500 332.1792 459.5205

Table 8 : Standard Deviation and Error Estimation for Norne Three Cases Standard Deviation

After Optimization Name Initial Case A Case B Case C Porosity (φ) 2.40929854 7.1731319 5.63125631 1.62107727

Permeability (k) 542.2171148 374.6854 372.6121 225.6877 Error Estimation

After Optimization Name Initial Case A Case B Case C Porosity (φ) 3.38637358 10.820077 7.43546765 1.60471051

Permeability (k) 663.8995657 359.7115 384.1416 199.5617 The above Tables 7 and 8 show the standard deviation and error estimation for Case 1 and all the three cases of Norne. For Case 1, after optimization, the standard deviation has increased from 0 to 7 and for permeability also it increased from 41 to 332 which are larger compared with the initial value. The average normalized error for the porosity is almost similar, only some fraction is higher than the initial value. Therefore, it is important

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to study the detailed differences between the estimated and real changes of the standard deviation and error measurement and that has given in Appendix 1. For Norne, considering porosity, the standard deviation of Case A and Case B give higher value compared to initial where Case C gives lower value than the initial. For permeability, Case A and Case B give lower value compared to initial, but Case C gives the lowest value compare with the values of Case A, Case B and the initial. For the error estimation, after optimization, in porosity, Case A and Case B show higher values than the initial value whereas Case C gives lowest value comparing to the initial. In permeability, all the three cases show lower value than the initial, but Case C shows the lowest.

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Chapter 7 Summary and Recommendations The improvement of model parameters such as porosity and permeability by assimilating data such as bottom hole pressure and production rates of oil, water and gas will be beneficial for field developments. It has been found that by using Ensemble Kalman Filter in a field implementation, a better history matched model can be achieved with improved porosity and permeability estimates. Using the updated fields and rerunning the model gives better match to the production data than the original history matched simulation. A good estimate of the current state of the reservoir is obtained at the end of model simulation. Updating of the model state with measurements gives the starting point for computing predictions. Thus, the methodology represents an ideal framework for reservoir monitoring and prediction. For parameter estimation, using Ensemble Kalman Filter is very efficient for updating reservoir models. The total CPU time for number of realizations in ensemble of reservoir models that match the latest production data is about the cost of running number of realizations in ensemble of reservoir simulations. Ensembles of reservoir models that are consistent with the up to date production data are always available for predictions for future performance with assessment of uncertainty. In this dissertation, results of one synthetic case (Case 1) and one semi-synthetic case of a real field data from Norne Field have been considered. The initial ensemble is made by using SPE comparative 3 for Case 1. For Norne 3 different initial ensembles are made where the initial ensemble for Case A and Case B of Norne field have been chosen randomly and for Case C, Sequential Gaussian Simulation (SGSIM) is used to prepare the initial ensemble.

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Some conclusion can be done from the results for Case 1 and Norne. From Figures (6.18 to 6.20) it can be concluded that, for Case 1, the EnKF shows small increase in porosity (30% instead of 20% in true condition) with decreased of standard deviation (0.06 instead of 0.1) around the production well. For permeability it gives the same value 1000 mD with same standard deviation (0.5) with respect to the true permeability. It can be said that EnKF plays an important role for parameter estimation. In Figure (6.21), the updated ensemble members at each time step as input for static fields give better match with the true measurement which indicates a better forecast using EnKF. In case of Norne (Figures 6.23 to 6.25), it can be concluded that, for three different cases, Case C gives better match compared to other two cases with less standard deviation for both porosity and permeability. For production (Figure 6.26 to 6.28), it can be summarized that Case A and Case B give better match with the true measurements for production of oil, water and gas with increased bottom hole pressure during production compared to true condition. On the contrary, Case C gives small reduction in the production of oil and gas with increased production of water in 3563 days where the bottom hole pressure for both the production and injection well remain same as the true condition. Considering the results of EnKF for each case, 100 ensembles’ porosity and permeability have been considered to further running the Eclipse and GLview software to construct the figures for real, initial and mean porosity for comparison. In parameter estimation, comparison of the real and mean porosity and permeability for the synthetic Case 1 shows (Figure 6.29 and 6.30) a difference for both the porosity and permeability with the real case. In Norne, from Figures (6.31 & 6.32), match with the real case, Case C gives better match both in porosity and permeability compared to Case A and Case B. For predicting future performance of the reservoir from Ensemble Kalman Filter, in Case 1, cumulative oil production (Figure 6.33) shows that, after assimilation the band of prediction increases compared to the predictions from the true and initial ensembles. It can be concluded that most of the realization members overpredict the cumulative oil production during the forecasting period might be the cause of some measurement noise. For Norne, considering the three cases, it can be said that all the cases have rising production behavior during the forecast period. Among them Case C gives very close match with the real measurement for most of the ensembles compared to Case A and Case B. Considering also the mean cumulative value for the production, and bottom hole pressure (Figures 6.45 to 6.48), Case C has a better match compared to Case A and Case B that gives an idea of consistent production forecast. It is obvious that the initial field with “correct” geostatistical information gives a better result. But it is not yet fully understood what important role is played by a correct prior geostatistical information in obtaining a best possible forecast.

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In objective function, Case 1 (Figure 6.51) shows the tendency of becoming stationary with the reduction values from the beginning after 13 time steps. It does not show the tendency of becoming zero due to some measurement errors which might be from the initial ensemble or from the filter performance. Whereas the comparison of the three cases for Norne (Figure 6.52) becoming nearly zero after 38 time step indicate that Case C shows less error which gives better results compared with Case A and Case B. Considering standard deviation and error estimation for Case 1 (Tables 7 & 8), after optimization, the porosity and permeability give higher values compared to the initial values for standard deviation. This gives the idea that the mean of the measurements is too far from the prediction. The average normalized error is almost similar for both porosity and permeability compared to the initial value which should be less for getting better results. Using an average number as diagnostic tools for evaluating the performance gives sometimes misleading result. For Norne, after optimization, Case C gives better results both in standard deviation and error estimation for porosity and permeability compared to the initial values and other two cases. From the table it is clear that Case C provides better results in standard deviation and error measurement compared to the other two cases because it gives minimum possible errors and standard deviation. From all the results above, it can be said that preparing initial ensemble by using Sequential Gaussian Simulation can give better results compared to the other two cases for both parameter estimation and optimization. However, for these small cases, the overall cost for 100 simulations for each case is small whereas large scale problems may be very demanding of computational resources. Hence, to reduce the number of ensemble members is a crucial issue for the EnKF. Therefore, carefully selection of initial members by using improved sampling algorithm is highly necessary. Recommendations:

To reveal heterogeneity in reservoir model with reduced uncertainty, assimilation of more production data is necessary. In some cases it is necessary to allow the updating of other variables in the assimilation using Ensemble Kalman Filter, such as transmissibility across fault, MULTZ, gas-oil and / or oil-water contact.

For improved prediction results, a relatively large ensemble size (200 or more

instead of 100) might be a better selection in order to adequately represent the uncertainty of model parameters.

New production well data during assimilation can further improve the estimation

quality of the reservoir model with further reduction of uncertainty resulting in more precise predictions.

An Ensemble Kalman Filter technology is developed for continuous updating of

reservoir simulation models where one synthetic and one semi-synthetic 2D case is

112

used. The methodology, however, can easily be extended to 3D case and can be applied to any existing reservoir simulator.

As the EnKF algorithm is excellent for parallel processing, real reservoir cases can

be implemented where several computers can be used for simulating the prediction steps.

To reveal more accurate results, 4D time-lapse seismic can also be used with

Ensemble Kalman Filter coupling with rock physics and seismic modeling software for parameter estimation and uncertainty prediction.

113

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Symbols

0A = ensemble matrix

kA = estimated states and parameters for all the ensembles C = linear measurement matrix Cα = covariance matrix which indicates amount of parameters correlations

dC = covariance matrix of the data D = measurements E = objective function

priorE = prior objective function

obsE = weighted sum of the squares of the differences between cald and obsd H = measurement matrix or Hessian matrix Hk = measurement index matrix at each time step k I = identity matrix K = Kalman gain matrix L = left factor for covariance matrix M = total number of measurements

TM = number of assimilation points Mn = number of measurements at timestep n N = ensemble size P = static parameters or covariance matrix for model uncertainty

0P = initial covariance matrix for the stochastic process x P10 = 10% quantile P50 = 50% quantile P90 = 90% quantile

fP = forecast using first ensemble

mP = forecast using mean ensemble

fPn = approximation to the model error covariance matrix Q = covariance matrix R = covariance matrix for measurement error U = state vector

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Ua = analysis step

fU = forecast step

fU n = forecast state vectors V = dynamic state variables W = diagonal matrix that assigns individual weights to each measurement X = random variable c = normalization constant ci = constraint function d = prediction quantity of interest

obsdr

= historical production data obsdi = measured data caldr

= calculated production data caldi = data calculated by the mathematical model of the reservoir

e 2.718281828≈ is called Euler’s number f = model function (reservoir simulator) ( )Xf x = probability density function for the continuous case

g( x ) = function of interest k = history match index / permeability j = measurement index nobs = number of observations (data)

ip = probability of X = xi for the discrete case q(z, t) = model noise q(z, tk) = model errors

ku = inputs at time k wi = weight factor for each data-point

x = input parameters like porosity, permeability ( ) ( )li 0 kx z ,t = states for ensemble l in grid block 0 at time k

kx = true state at time k

k 1x−+ = a priori estimate of the process state

kx = estimate of the system state with filtered information up to time tk lx = ensemble matrix lx = ensemble mean

x− = a priori estimate y = predicted quantity like BHP, RS, RV, OPR with time

ky = measurements at time k

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*y = available measurements up to time tk

ly = true measurement at space l tjy = "true" measurement from the original process

jY = measurement matrix

jε = measurement perturbation matrix 2σ = variance

σ i = standard deviation of measurement error of thi observation φ = porosity α = reservoir model parameters

priorα = priori mean value of the means, variances and covariance of the parameters

μ = shear modulus of solid framework *μ = weighting factor which expresses the relative strength in the initial model ( ) ( )lj 0 kz ,tq = poorly known parameters to be estimated

w = real number Φ = linear model matrix ∆ = input matrix

Subscripts f = forecast (a priori)

a = analyzed (a posteriori) m = model noise o = observation t = true h = horizontal / history matched v = vertical i = index that runs over all points in the dataset

Superscripts T = matrix transpose n = timestep index i = ensemble member index − = a priori estimate . = ensemble mean

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Nomenclature BHP = Bottom Hole Pressure COSGSIM = Sequential Gaussian Co-simulation COSISIM = Sequential Indicator Co-simulation DSSIM = Direct Sequential Simulation EKF = Extended Kalman Filter EnKF = Ensemble Kalman Filter GRF = Gaussian Random Field IK = Indicator Kriging KF = Linear Kalman filter KT = Kriging with a trend MAP = Maximum a Posteriori MCMC = Markov Chain Monte Carlo method OK = Ordinary Kriging SGSIM = Sequential Gaussian Simulation SISIM = Sequential Indicator Simulation SK = Simple Kriging RF = Random Function RMS = Risk Management Solution RS = Initial gas-oil ratio RV = Random Variable RV = Initial Vapor oil-gas ratio cdf = Cumulative Distribution Function ccdf = Conditional Cumulative Distribution Function mp = Multiple Point nst = Normal Score Transform pdf = Probability Density Function

Dr = Perturbation parameter

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Appendix A Table 9: Standard Deviation and Objective Function Case 1:

Real Initial Ensembles Poro (φ)

Perm (k)

Poro (φ)

Perm (k) # Porosity (φ) Permeability (k)

SD Error SD Error SD Error SD Error SD Error SD Error 0 0 0 0 0.068 10.067 41.031 500 1 8.421169 11.21616 418.3403 497.8048

2 7.907622 12.10104 376.0212 457.5361 3 7.628853 11.23678 395.7013 487.3183 4 8.741659 12.90992 368.5013 481.0862 5 9.451047 12.7674 413.5353 473.4405 6 9.047754 11.6169 546.3385 577.1992 7 8.111153 11.51032 421.6787 491.8217 8 8.349435 11.69271 377.2323 463.5612 9 7.349035 11.73332 349.786 474.9209 10 7.69602 11.10997 376.8591 449.5605 11 7.401532 10.70936 540.7238 496.2146 12 8.115684 12.07301 514.074 556.4846 13 7.986534 11.4517 348.5932 440.3395 14 7.658247 12.11923 512.2699 500.3413 15 8.065174 11.5781 353.4663 435.5143 16 8.253343 11.8631 473.509 530.0128 17 7.717889 11.97195 433.0948 472.2871 18 9.061431 11.66318 460.439 524.1192 19 8.020507 11.65596 515.9543 476.4775 20 7.651885 10.5537 387.6641 448.9378 21 8.83045 11.99142 494.7282 513.0287 22 9.102143 12.21617 452.0259 549.2227 23 8.280119 11.17622 425.034 509.5447 24 8.127796 10.8956 376.6163 440.3854 25 7.971641 11.1382 422.4336 526.7441 26 8.670739 11.6846 370.5715 464.8086 27 8.095616 11.50695 441.5008 466.8434 28 7.728083 10.85268 753.0868 529.5134 29 8.512122 11.50203 379.5592 456.0866 30 8.345458 12.69265 528.1539 587.3473 31 8.323641 11.94206 422.8078 516.6516 32 9.009938 12.30975 425.3724 517.6189 33 9.276779 12.73889 423.312 527.2271 34 7.818171 11.66178 431.3746 639.6688 35 8.786982 11.72401 367.0942 501.2268 36 8.521988 11.76415 487.1853 542.5767 37 9.263225 12.43597 400.1198 466.3675 38 8.25541 12.05472 348.5214 438.1159 39 8.787304 11.92038 398.8607 495.354 40 7.593668 10.03575 443.2055 548.429 41 7.617325 11.77383 402.8409 533.8615 42 8.308488 12.73328 477.54849 490.1532 43 8.298751 11.99668 405.7114 540.4878

128

44 8.376548 11.44275 379.7965 481.0406 45 7.908149 11.46044 410.8228 484.4139 46 7.787566 10.87335 345.2608 469.0087 47 8.483761 12.49549 412.8164 525.6078 48 8.474819 12.99114 417.3262 482.115 49 8.0015 11.11952 473.9104 521.5169 50 8.737579 11.89939 394.4439 461.5719 51 8.261109 11.68967 418.7524 513.353 52 9.036336 13.35032 492.9047 533.3788 53 9.903609 13.46826 476.1371 485.1726 54 7.576954 11.16655 348.0772 452.5666 55 9.012206 11.90463 375.0696 461.669 56 9.048625 12.19009 588.5679 499.5219 57 8.200663 11.02245 404.5642 567.1838 58 8.805787 12.47739 422.6451 494.8511 59 8.400187 11.23649 375.2062 472.2657 60 7.952581 11.16818 338.1415 466.5697 61 8.45454 11.93723 432.4541 488.0564 62 8.912612 12.17553 423.1232 444.8962 63 8.451175 11.54704 385.6219 466.2924 64 8.845538 12.87868 565.249 520.048 65 7.242531 10.51411 479.7721 538.8127 66 8.227862 11.55545 349.7249 448.894 67 8.663935 11.49115 373.1264 517.569 68 8.381589 11.48064 415.6952 503.8353 69 9.061518 13.25302 488.3181 520.1271 70 8.721184 12.65999 419.5119 486.7903 71 8.496641 11.14489 369.6084 454.5986 72 8.422547 11.74008 379.7838 479.3811 73 8.696502 12.02523 431.5008 478.8407 74 7.627502 10.78345 381.25 487.2964 75 7.114391 11.14368 374.305 479.6111 76 8.998199 12.74376 513.1036 511.9622 77 8.750519 12.15993 372.2388 466.2613 78 8.590518 11.35054 332.6622 451.7336 79 9.439436 12.82505 401.4225 490.345 80 8.267068 12.04655 433.3856 544.5263 81 9.267751 12.76866 450.2017 578.2053 82 8.13493 11.83479 471.9018 582.4856 83 8.23885 11.37541 391.1916 476.114 84 8.527319 13.2433 440.1356 562.2904 85 8.843384 12.54343 400.6335 467.3337 86 7.430619 11.61235 450.827 537.5489 87 8.580362 12.07494 409.2314 516.8096 88 7.201676 10.82991 391.1558 509.4163 89 7.203218 12.45664 390.4493 501.4804 90 8.780247 12.38091 386.89 540.1182 91 8.373782 11.5249 592.0983 571.7937 92 8.042041 11.32695 444.9341 504.0135 93 8.154123 11.36228 336.9964 444.0594 94 8.30929 11.57883 425.8395 563.8797 95 9.219218 12.02889 437.11 497.4062 96 8.797847 12.63124 454.5295 522.9016 97 8.335865 12.1219 493.9804 531.1684 98 9.286239 12.83803 419.115 478.7511 99 7.751602 11.40719 519.3542 537.161 100 9.607648 12.33835 397.0519 457.0794

129

Table 10: Standard Deviation and Objective Function

Norne Standard Deviation

Real Initial Ensembles Case A Case B Case C Poro

(φ) Perm (K)

Poro (φ)

Perm (K) # φ K φ K φ K

0.90112 49.95026 2.409295 542.2171 1 7.8958 413.2996 6.509108 390.4533 1.851232 295.9369 2 7.961211 411.7924 7.101608 434.5172 1.861246 357.2122 3 7.939011 403.3636 6.511796 390.1202 1.949881 292.9956 4 8.887626 400.6073 7.048795 427.6275 2.017892 318.2189 5 7.819992 414.9381 6.928205 405.0581 1.768563 276.5743 6 9.210068 429.7037 7.088754 370.0438 1.915459 263.5804 7 7.964692 395.7997 6.659889 403.9707 1.640801 413.2477 8 8.693757 375.8925 6.433076 394.9044 2.208533 473.7007 9 8.085406 509.0554 6.529031 390.0558 1.696248 252.5367 10 7.772283 404.5602 6.944216 396.5191 1.944274 310.536 11 8.155442 380.7829 6.925268 406.1363 1.902068 409.3259 12 7.235711 376.9425 7.046444 344.6469 1.722884 343.5892 13 8.536696 422.2268 6.757387 420.7361 1.950532 572.3516 14 8.306055 384.1254 5.946579 421.5681 1.829004 306.0199 15 7.928079 429.1442 6.97752 400.2854 1.631363 425.7428 16 7.898765 506.5787 6.948276 376.4686 1.909288 568.0524 17 9.089736 401.2317 7.212812 388.9593 1.777153 293.5226 18 8.042653 425.3754 5.743118 380.3681 1.969319 442.2294 19 7.94974 362.8563 6.544146 415.7474 2.274512 313.786 20 7.724651 415.0486 7.145078 401.5121 1.948121 352.9282 21 8.641642 435.5541 7.916757 400.5532 1.886435 314.95 22 7.315696 412.2156 7.616319 386.7606 1.987644 351.9032 23 7.699814 399.1298 7.484089 436.5075 2.018583 348.3059 24 8.238836 484.8415 7.208624 422.4432 1.771967 322.7116 25 7.30387 451.2961 6.179052 396.9153 1.86503 342.6445 26 8.743422 403.0017 7.176985 422.8733 1.939567 325.6282 27 7.728645 367.4818 6.791742 413.9317 1.934044 741.834 28 7.693906 340.8138 5.939167 412.6823 1.507755 256.4462 29 8.640963 436.3377 6.877121 423.4638 1.984351 359.3511 30 8.144346 414.6096 7.198651 407.2769 2.014748 315.0199 31 9.091886 406.1215 7.42492 384.6532 1.929367 364.7328 32 8.17411 415.9036 6.888623 397.5463 2.096292 339.3294 33 7.828361 409.8591 7.645111 409.2042 2.021052 614.906 34 8.278817 435.816 5.599956 377.1571 1.841492 229.2485 35 8.439556 385.1726 6.988599 408.1614 1.780635 444.1252 36 8.023146 417.8657 8.037152 386.7587 1.623031 554.7876 37 7.472941 370.0986 7.616286 477.2339 1.814736 361.1002 38 7.757086 400.1095 7.846264 395.0062 1.783767 278.9694 39 8.105371 389.5724 6.886011 406.5759 1.751704 257.579 40 8.280045 395.3894 6.417841 394.2554 1.756895 300.249 41 8.945324 350.3093 6.733721 409.859 1.906398 489.1662 42 8.851161 399.1914 8.041828 434.8751 2.090434 324.5252 43 7.870255 424.7936 6.645857 350.0039 1.892251 296.098 44 8.996655 431.9139 7.596942 406.3615 1.739335 264.8232 45 9.199844 449.98 7.132238 412.4521 1.678145 445.8293 46 8.872808 431.9296 7.688709 372.5359 1.91054 324.672 47 8.480197 441.2725 7.471464 419.5304 1.907844 278.4329 48 8.564631 394.4463 7.359529 398.3375 1.965142 1136.615 49 8.322788 406.3432 7.04327 414.9172 1.727769 366.4602

130

50 7.702393 466.249 7.856825 392.3255 2.067247 385.424 51 8.165455 378.7477 7.473094 371.7125 1.907969 541.4087 52 8.336579 386.9254 6.316422 373.4979 1.672675 326.0717 53 9.073604 370.077 6.951874 397.2367 1.70283 361.9907 54 8.117939 390.2375 7.040609 392.1641 1.753849 296.4996 55 8.303432 366.9074 6.571616 465.7428 1.662629 284.9774 56 7.856831 450.5966 6.84391 392.1771 1.764988 428.5688 57 7.752023 366.801 5.882572 384.112 2.016285 301.6297 58 8.653693 414.8551 7.818897 437.8295 1.75868 463.8758 59 9.732449 421.1007 7.099441 425.6709 1.904326 311.0767 60 7.799498 441.4872 7.429236 353.2073 1.87221 631.3191 61 7.768523 417.9206 6.705116 376.3104 1.97662 415.3017 62 8.229795 410.1807 6.66252 380.8677 1.840358 265.3652 63 8.792054 380.6771 7.213313 436.8151 1.776059 314.187 64 7.756198 381.7642 6.913062 464.4784 1.950198 398.3781 65 8.850464 407.8574 6.946008 431.2562 1.660302 274.5735 66 8.020498 378.2736 7.311319 425.9589 2.055173 323.6769 67 8.265625 386.5305 6.625227 361.2219 1.800487 281.3173 68 8.485231 391.2419 6.108506 372.7361 2.193719 312.1311 69 7.876299 397.1516 6.859075 392.2371 1.814543 223.3652 70 8.867627 442.4612 6.659206 450.664 2.050555 261.7461 71 8.368509 417.8238 7.351839 404.7985 1.8614 687.5412 72 7.367052 423.2049 6.4077 399.2881 1.895052 298.5091 73 7.965954 368.5155 6.860227 357.2309 1.813791 260.7606 74 7.811908 376.2082 6.701791 370.9683 2.025852 458.1848 75 8.670635 404.9926 6.948947 445.1089 1.896254 312.6602 76 7.342355 378.6401 7.514344 392.6729 2.110908 525.1147 77 8.54462 423.3795 7.165922 398.6214 1.83724 413.1952 78 7.748928 379.6272 6.304558 390.0904 2.029078 383.9706 79 9.280388 404.0047 8.269411 429.5603 2.065373 305.0695 80 8.308102 396.7905 6.005085 380.8915 1.97261 272.3484 81 7.754594 421.6802 6.58776 366.0219 2.019903 269.5221 82 7.067154 396.1352 7.439852 387.5062 2.081372 321.0954 83 7.714029 441.2675 7.644138 379.4298 1.924318 283.4158 84 8.657311 451.4321 7.056981 405.5835 1.967676 337.3389 85 7.758272 442.6989 6.613495 396.3652 1.890434 345.5696 86 8.61684 390.1158 6.82798 364.2167 1.947747 306.2584 87 8.225364 350.2743 6.629843 394.1178 1.930377 362.3223 88 8.152533 395.3748 6.828062 397.2904 1.741464 280.284 89 8.012971 392.6917 7.019941 390.3146 2.018823 338.3467 90 7.83217 405.5107 7.390205 432.8246 1.768409 296.2402 91 8.992668 394.2764 5.71999 428.4509 1.806426 385.7447 92 7.805104 412.119 6.45207 394.902 2.082838 312.2687 93 7.953295 411.7811 6.809969 402.3105 2.141952 631.1111 94 7.487627 359.4331 6.282662 361.3692 1.828664 346.478 95 8.135188 405.3568 7.5356 389.4963 1.925103 267.837 96 8.100005 436.57 7.202652 422.5789 1.869883 344.7371 97 9.815441 439.9004 7.940199 376.9869 1.907869 336.136 98 7.726703 394.4176 6.69572 392.9962 1.930637 338.7388 99 8.040149 388.3642 6.432716 389.5074 1.80861 281.2295 100 7.851548 377.4285 6.287725 420.3001 1.750566 674.0612

131

Table 11: Standard Deviation and Objective Function Norne Error Estimation

Real Initial Ensembles Case A Case B Case C Poro

(φ) Perm (K)

Poro (φ) Perm(K) # φ K φ K φ K

0.04596 9.634044 3.386374 663.89957 1 11.72699 370.6804 8.981338 409.7477 1.961072 244.7428 2 12.06944 402.3068 10.17766 439.6328 2.160664 281.4687 3 11.75941 404.4751 10.45687 395.4596 2.099046 251.5425 4 12.09576 381.7762 10.051 397.4927 2.399877 284.2021 5 11.67863 378.2623 9.918446 389.9148 1.806194 208.0673 6 14.01027 393.1771 9.745188 373.2923 1.975283 231.5087 7 12.50023 378.5069 9.706499 428.4599 1.800318 236.8725 8 12.28474 354.7378 8.56096 405.0984 2.455832 341.6043 9 11.98783 394.0296 9.589821 389.8889 1.705979 203.2334 10 12.87061 380.8687 9.270424 418.9135 2.117796 262.7693 11 12.31659 375.9314 8.572191 404.6901 2.186973 290.4624 12 11.08263 348.3875 9.817262 378.5535 1.941756 245.2227 13 11.73185 378.9718 8.146184 409.1719 2.063266 320.4809 14 13.36322 379.082 9.383784 419.6888 1.935856 234.6102 15 11.60738 363.6247 9.28381 406.256 1.789934 268.1073 16 11.42559 392.9606 10.56185 404.2613 2.216337 327.9679 17 13.08472 376.7979 9.735637 381.3757 1.856213 232.4417 18 11.688 359.5845 7.529568 379.2411 1.962313 287.0994 19 11.59108 380.5956 10.0786 404.3801 2.164552 265.2418 20 11.8674 373.6575 9.853986 416.5621 2.084762 266.8905 21 12.57713 414.0761 11.04596 407.4602 2.086378 259.3535 22 11.80422 402.7695 9.523915 376.2159 2.231337 286.269 23 12.21568 363.4282 11.59228 432.5875 2.17152 263.3472 24 12.03072 413.2357 10.52111 417.5373 1.888227 251.6983 25 11.8253 430.7785 9.321156 375.7806 1.917022 263.2488 26 12.94307 389.1164 8.374269 392.4219 2.197838 275.7681 27 11.29694 372.3405 9.554385 421.3698 2.153869 341.2872 28 12.10048 348.0258 7.555745 400.9247 1.59831 210.0869 29 11.97668 388.8691 10.30119 420.0266 2.040915 253.3449 30 11.4717 379.6667 9.408929 415.1665 2.27204 270.9743 31 13.39606 382.0772 9.500496 407.1693 2.032343 278.2075 32 12.19235 386.2504 9.225595 389.0005 2.204503 276.671 33 12.08701 387.8277 9.451284 419.0769 2.159007 302.4967 34 11.84045 382.7355 7.451281 380.9115 1.965001 191.8293 35 12.71053 378.3705 9.562948 427.9311 1.795251 262.6989 36 11.99295 381.396 9.75534 391.6692 1.771203 277.8849 37 11.51494 375.7438 9.156297 430.5089 1.884084 249.3859 38 11.28461 387.4023 11.76752 406.1023 1.895926 232.4299 39 13.02474 369.0412 9.336593 413.4303 1.839298 218.1345 40 12.62681 375.7981 8.731662 368.3138 1.861594 225.513 41 12.84746 348.6326 11.20514 439.5967 2.055718 291.1341 42 13.37883 367.7458 11.547 466.78 2.142338 276.105 43 11.30002 389.5132 8.555736 349.11 1.944339 233.05 44 13.40939 429.023 11.74221 433.0779 1.882369 217.3282 45 13.44138 422.0299 9.925532 433.5995 1.762986 269.4253 46 13.46558 405.5464 11.45288 378.4453 2.162845 262.0092 47 12.91543 394.8703 10.74671 386.4455 2.068282 233.4029 48 12.44794 370.2266 9.662613 428.1209 2.244042 368.5344 49 13.59517 374.2343 8.913106 419.8766 2.136906 288.5654 50 11.64929 381.7054 10.49517 399.6827 1.900926 266.8333

132

51 12.27591 379.4211 10.97275 357.5846 2.389239 324.2502 52 11.88297 356.403 9.631154 367.6224 1.864348 245.0316 53 12.54901 365.2794 10.56667 365.0795 2.062121 261.4181 54 12.45226 364.7719 9.66446 394.3004 1.91274 236.6531 55 11.98455 368.6296 9.02348 430.9832 1.87593 242.6415 56 12.06814 387.0529 9.147856 399.1918 1.962207 273.4255 57 11.11788 359.1197 7.671773 375.0546 2.033264 250.4409 58 12.83189 395.2658 9.469768 420.0409 2.068153 300.7998 59 12.64584 389.1383 9.549931 411.7209 1.878612 241.837 60 12.09549 394.8092 10.39477 360.182 2.014541 285.364 61 12.31757 380.4938 9.333919 375.9265 2.235682 253.7244 62 12.25641 394.3855 10.3117 388.4647 2.085419 242.2726 63 12.63434 371.7161 10.70438 418.7712 2.053982 252.3854 64 11.57326 354.1706 9.275372 465.4228 2.155914 256.1587 65 13.46363 401.8743 10.38267 387.5592 1.89783 217.7714 66 10.61658 382.9353 10.92359 438.8724 2.122273 362.9358 67 12.70988 379.4769 9.160301 354.47 1.956955 244.6711 68 11.53748 377.1162 9.585934 386.3854 2.361238 280.3532 69 12.14748 353.1538 10.81192 410.8969 1.941504 202.5769 70 13.82632 383.3504 9.638863 454.0254 2.097795 225.915 71 13.18362 374.344 8.658741 410.7274 2.057235 297.6454 72 11.62753 381.441 8.254674 408.3093 2.02226 241.6826 73 11.41761 373.439 9.170111 356.1207 1.986503 235.8598 74 12.31886 388.8267 8.935421 366.0885 2.213452 306.8956 75 12.62138 376.9332 9.950697 440.1402 2.024096 242.5661 76 10.35123 370.1402 10.41956 381.5914 2.132242 279.5324 77 12.18029 400.4719 9.743435 397.7728 2.004046 274.712 78 11.19069 380.0188 8.805174 376.1647 2.044864 277.0082 79 13.07799 388.8439 9.723709 412.8386 2.304689 236.4311 80 11.57259 353.7889 7.566547 366.8947 2.040259 235.0317 81 11.29664 372.7154 7.480915 377.9918 2.212429 238.2546 82 11.84652 370.506 10.53802 427.8498 2.070298 251.3179 83 12.02836 399.5899 9.568011 395.3143 2.096862 236.582 84 13.28105 373.1168 10.06116 408.8071 2.045698 271.2861 85 12.12917 395.4608 8.942185 359.1507 2.1444 236.2706 86 12.71711 360.3931 8.374179 396.1711 2.091882 251.5146 87 12.16214 358.431 8.814487 415.3943 2.077585 278.9575 88 11.67618 375.6729 9.195287 386.8486 1.965238 224.0103 89 11.86124 357.3746 9.052808 371.4039 2.237784 244.2108 90 12.15774 394.1947 8.891557 418.3346 1.928203 248.487 91 12.66278 394.22 7.181739 388.5551 2.082679 284.1891 92 11.70497 369.4795 9.28517 391.2885 2.155289 251.1885 93 11.47136 382.3137 9.548042 379.7493 2.313361 329.9184 94 10.48617 370.1873 8.916431 363.506 1.946688 267.9548 95 11.5922 400.2366 10.87814 379.0888 2.07442 238.4686 96 12.75525 386.3588 8.945726 431.1601 1.904761 232.7664 97 13.28512 404.7877 9.011206 376.3202 2.223008 277.3342 98 11.05154 389.0653 9.677312 398.1572 2.181397 253.6837 99 11.77478 381.5749 9.650578 3.858421 1.780883 233.8069 100 11.04248 354.6919 8.796798 395.0168 1.886414 266.6721

133

Appendix B Table: Real Error Ensembles Case 1 Norne

Case A Case B Case C 1 4.60E-02 1.01E-02 1.85E-03 1.26E-03

2 5.01E-02 4.25E-03 2.71E-03 8.10E-04

3 6.19E-02 4.81E-03 5.53E-03 1.06E-03

4 2.06E-02 6.13E-03 7.41E-03 3.64E-03

5 9.18E-02 8.10E-03 1.14E-02 1.31E-03

6 5.80E-02 6.13E-03 1.61E-03 1.37E-03

7 6.09E-02 4.59E-03 6.30E-03 3.04E-04

8 5.16E-02 4.41E-03 5.96E-03 5.60E-04

9 7.16E-02 1.40E-03 1.18E-02 2.51E-03

10 5.38E-02 5.26E-03 2.85E-03 2.16E-03

11 3.39E-02 6.24E-03 6.91E-03 1.41E-03

12 5.13E-02 4.87E-03 8.62E-03 2.97E-03

13 6.31E-02 5.83E-03 7.50E-03 1.23E-03

14 6.94E-02 3.23E-03 1.13E-02 3.08E-03

15 3.69E-02 3.00E-03 2.19E-03 1.28E-03

16 3.66E-02 9.60E-03 5.31E-03 1.30E-03

17 2.82E-02 5.43E-03 4.94E-03 1.51E-03

18 6.73E-02 9.31E-03 4.79E-03 7.72E-04

19 2.87E-02 8.52E-03 8.21E-03 1.33E-03

20 4.11E-02 3.01E-03 5.42E-03 1.45E-03

21 6.19E-02 2.96E-03 4.44E-03 1.14E-03

22 3.33E-02 5.33E-03 1.50E-03 4.86E-03

23 4.58E-02 1.14E-02 1.70E-03 2.24E-03

24 4.66E-02 5.43E-03 5.62E-03 1.96E-03

25 3.77E-02 5.08E-03 6.04E-03 8.09E-03

26 3.07E-02 1.02E-02 2.92E-03 1.24E-03

27 2.97E-02 9.06E-03 6.97E-03 2.37E-03

28 5.36E-02 7.74E-03 2.84E-03 1.07E-03

29 2.75E-02 4.93E-03 2.66E-03 1.16E-03

30 6.51E-02 4.01E-03 8.24E-03 3.85E-03

31 0.1088383496 7.76E-03 3.47E-03 2.51E-03

134

32 6.27E-02 7.44E-03 4.55E-03 4.04E-04

33 8.26E-02 6.94E-03 4.18E-03 1.26E-03

34 2.65E-02 3.29E-03 4.11E-03 1.09E-03

35 3.72E-02 5.98E-03 4.32E-03 1.61E-03

36 5.81E-02 6.28E-03 1.12E-03 8.18E-04

37 3.45E-02 3.68E-03 1.13E-03 3.46E-03

38 2.81E-02 1.44E-02 2.40E-03 1.39E-03

39 8.41E-02 7.46E-03 2.09E-03 1.31E-03

40 5.60E-02 5.65E-03 9.15E-03 1.41E-03

41 3.00E-02 4.56E-03 7.44E-03 8.30E-03

42 3.44E-02 3.74E-03 3.76E-03 1.03E-03

43 3.56E-02 9.74E-03 3.61E-03 3.15E-03

44 6.42E-02 1.10E-02 4.13E-03 9.52E-04

45 2.60E-02 5.76E-03 1.82E-03 5.26E-04

46 5.06E-02 2.70E-03 2.24E-03 4.57E-03

47 2.70E-02 7.41E-03 7.25E-03 1.53E-03

48 5.54E-02 3.31E-03 6.54E-03 2.64E-03

49 6.99E-02 7.67E-03 3.02E-03 2.29E-03

50 3.87E-02 1.66E-02 7.63E-03 4.06E-03

51 5.35E-02 1.09E-02 4.83E-03 1.78E-03

52 2.78E-02 7.53E-03 8.52E-03 2.92E-03

53 7.53E-02 5.87E-03 6.19E-03 8.85E-04

54 2.50E-02 7.57E-03 5.19E-03 1.15E-03

55 4.32E-02 9.82E-03 4.49E-03 1.39E-03

56 5.47E-02 1.24E-02 2.97E-03 1.07E-03

57 4.13E-02 3.44E-03 5.05E-03 3.39E-03

58 6.35E-02 6.80E-03 3.07E-03 5.25E-03

59 3.93E-02 8.13E-03 5.83E-03 1.11E-03

60 4.10E-02 1.12E-02 1.15E-02 2.34E-03

61 9.64E-02 3.76E-03 9.03E-03 6.61E-03

62 2.20E-02 4.92E-03 8.37E-03 1.26E-03

63 3.20E-02 9.72E-03 6.71E-03 8.86E-04

64 7.68E-02 4.01E-03 5.55E-03 1.93E-03

65 3.35E-02 5.68E-03 6.23E-03 3.05E-03

66 4.69E-02 4.44E-03 3.97E-03 2.46E-03

67 5.66E-02 1.04E-02 2.52E-03 2.50E-03

68 6.21E-02 4.86E-03 3.56E-03 2.70E-03

69 3.68E-02 5.72E-03 3.52E-03 1.58E-03

70 3.32E-02 5.79E-03 7.34E-03 4.19E-03

135

71 7.56E-02 3.68E-03 3.74E-03 1.44E-03

72 4.39E-02 1.19E-02 1.87E-03 2.07E-03

73 3.85E-02 6.64E-03 5.19E-03 1.73E-03

74 4.86E-02 5.61E-03 4.85E-03 1.03E-02

75 5.92E-02 4.55E-03 2.78E-03 1.22E-03

76 5.94E-02 9.00E-03 5.31E-03 8.33E-04

77 6.02E-02 5.86E-03 4.82E-03 1.39E-03

78 1.86E-02 5.02E-03 7.69E-03 2.62E-03

79 2.69E-02 3.49E-03 7.17E-03 1.37E-03

80 4.00E-02 7.61E-03 3.36E-03 2.34E-03

81 3.41E-02 5.64E-03 7.60E-03 1.60E-03

82 3.35E-02 4.96E-03 5.91E-03 2.85E-03

83 3.06E-02 4.05E-03 3.09E-03 8.89E-04

84 3.19E-02 6.26E-03 2.53E-03 1.01E-03

85 3.64E-02 1.13E-02 4.92E-03 6.38E-04

86 7.03E-02 5.17E-03 4.33E-03 3.42E-03

87 6.30E-02 6.77E-03 5.58E-03 4.29E-03

88 4.63E-02 2.63E-03 4.14E-03 6.05E-03

89 6.61E-02 5.74E-03 4.09E-03 4.31E-03

90 3.41E-02 3.16E-03 3.11E-03 2.49E-03

91 4.01E-02 5.51E-03 5.52E-03 1.29E-03

92 0.1113625318 6.22E-03 7.28E-03 1.24E-03

93 4.00E-02 5.20E-03 3.73E-03 1.39E-03

94 6.22E-02 5.66E-03 2.72E-03 1.87E-03

95 6.22E-02 5.05E-03 3.11E-03 8.52E-04

96 3.36E-02 2.91E-03 5.73E-03 5.22E-04

97 6.53E-02 4.12E-03 3.82E-03 1.52E-03

98 4.03E-02 7.00E-03 9.00E-03 1.09E-03

99 2.79E-02 4.04E-03 1.40E-02 2.55E-03

100 2.45E-02 4.87E-03 9.31E-03 5.64E-03