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ECMOR XII - 12th European Conference on the Mathematics of Oil Recovery
- Conference date: 06 Sep 2010 - 09 Sep 2010
- Location: Oxford, UK
- ISBN: 978-90-73781-89-4
- Published: 06 September 2010
81 - 100 of 117 results
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An Iterative Scheme to Construct Robust Proxy Models
Authors A. Castellini, H. Gross, Y. Zhou, J. He and W. ChenQuantifying and understanding uncertainty in reservoir production forecasts is the key to robust reservoir management decisions. Monte-Carlo simulation is generally impractical because of the large number of subsurface realizations and the computationally intensive flow simulations. Response surface models have been introduced to improve the efficiency of the traditional asset development workflows: uncertainty assessment, history-matching and optimization of development plan. Linear regression techniques are the most popular methods to create the analytical response surfaces. However, the resulting proxies can be poor predictors of reservoir performance when strong non-linear effects are significant. We are proposing an iterative sampling strategy that is able to capture the non-linear behavior of the response and efficiently refine the proxy model. Thin-plate spline non-linear regression techniques have been selected to construct proxy models as they present a number of attractive properties. At each iteration, new combinations of parameters are rapidly evaluated with a score function and the best ones are selected for flow simulation. The benefit of the iterative scheme is demonstrated on a new field development and a mature asset with historical data. For a fixed computational cost, iterative response surfaces consistently provide better accuracy than traditional designs
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A Comparison of Screening Methods for High Dimensional Input Problems
Authors S. Touzani, D. Busby and A. AntoniadisHistory matching studies can involve adjusting a high number of reservoir uncertain parameters. Sensitivity analysis can help reservoir engineers focusing on the most influential parameters to prioritize the effort thus reducing sensibly the history matching time. However, due to nonlinear and interactions effects and depending on the method used, this exercise can be misleading or in some cases very time consuming. In this work, several statistical nonparametric methods from the most recent to the most classical have been implemented and tested on some typical oil reservoir applications. Different tests are made using different problems of different dimension. Most of the nonparametric methods investigated are based on response surfaces such as kriging, polynomial or smoothing splines responses with different types of penalization (LASSO, COSSO,...), and some others are issued from the recent statistical literature. The response surface methods were tested using space filling experimental design such as maximin latin hypercube. In the comparisons the computational cpu time for each method is also reported because even if these are generally negligible respect to large simulation time, they can be considerable for large dimensional input problems. To assess the quality of the methods several criteria have been used such as the prediction accuracy of the response surfaces, but also other criteria such as the number of times the method fails to detect an influential parameter (type 1 errors) or the number of times it indicates a non influential parameter as influential (type 2 errors). Numerical tests were made on different outputs (objective functions) of realistic synthetic reservoir models.
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Parametrized Model Order Reduction Applied to Reservoir Simulation
By E. GildinThe development of efficient numerical reservoir simulation is an essential step in devising advanced production optimization strategies and uncertainty quantification methods applied to porous media flow. In this case, a highly accurate and detailed description of the underlying models lead to a solution of a set of partial differential equations, which after discretization, induce dynamical systems of very large dimensions. In order to overcome the computational costs associated with these large-scale models, several forms of model-order reduction have been proposed in the literature. In porous media flow, two different approaches are used: (1) a "coarsening" of the discretization grid in a process called upscaling; and (2) a reduction in the number of state variables (i.e., pressure and saturation) directly in a process called approximation of dynamical systems. Recently, the the idea of combining both approaches have been proposed using the control-relevant upscaling (CRU) methodology. In this paper, we investigate the use of the so-called parametric model order reduction (PMOR) techniques applied to porous media flow simulation in a system-theoretical framework. PMOR entails the generation of reduced-order models which retains the functional dependency on specific parameters of the original large-scale system. In particular, this work focuses on the the application of PMOR to the case of single-phase flow, in which the dependencies of the porous media properties, such as, permeability, and the discretization parameter, such as, grid sizes, is investigated. The the main ideas behind model order reduction will be reviewed, including the general framework of interpolatory projection techniques and applications to single-phase flow test cases will be developed.
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Application of Inflow Control Devices to Heterogeneous Reservoirs
Authors V.M. Birchenko, A.I. Bejan, A.V. Usnich and D.R. DaviesThe rate of inflow to a long well can vary along its completion length, e.g. due to frictional pressure losses or reservoir heterogeneity. These variations often negatively affect the oil sweep efficiency and the ultimate oil recovery. Inflow Control Devices (ICDs) represent a mature well completion technology which provides uniformity of the inflow profile by restricting high specific inflow segments while increasing inflow from low productivity segments. This paper introduces a mathematical model for effective reduction of the inflow imbalance caused by reservoir heterogeneity. The model addresses one of the key aspects of the ICD technology application - the trade-off between well productivity and inflow equalisation. Our analytical model relates the specific inflow rate and specific productivity index to well characteristics taking into account the intrinsically stochastic nature of reservoir properties along the well completion interval. A general solution to our model is available in a non-closed, analytical form. We have derived a closed form solution for some particular cases. The practical utility of the model is illustrated by considering a case study with prolific and medium productivity reservoirs. Finally, we identify limitations in using our model.
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An Expanded Well Model for Accurate Simulation of Reservoir-well Interactions
Authors M. Karimi-Fard and L.J. DurlofskyWe present a new framework for modeling wells in reservoir simulation. This approach is based on an “expanded well model” in which the well region is expanded geometrically to include portions of the reservoir. The well region is then represented as an “equivalent” multi-segmented well defined by a list of connections for all completed segments. This generalizes and simplifies well-reservoir flow modeling by shifting the interface between the well region and the reservoir region. As is the case with standard well models, the well region is linked to the reservoir region through use of well indices. In the case of the expanded well model, these well indices, along with the connections between segments in the equivalent multi-segmented well, are computed by upscaling an underlying fine-scale description of the reservoir and well. The method is applied to model a hydraulically-fractured well and production in a tight-gas reservoir. In both cases, direct application of standard approaches is shown to lead to inaccurate coarse-scale predictions. Use of the expanded well model, by contrast, provides high degrees of accuracy in both cases.
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Modeling of the Heel–toe Effect in a Horizontal Well with Inflow Control Devices
Authors V.M. Birchenko, K.M. Muradov and D.R. DaviesThe specific inflow rate of a horizontal well normally varies along the well's completion length due to either frictional pressure losses in the well (heel-toe effect) or variation in the well’s specific productivity index and pressure support. Downhole Inflow Control Devices (ICDs) capable of exerting an additional, rate-dependant restriction along the completion can reduce such inflow imbalances. This paper discusses a new method to quantify the pressure losses and inflow distribution along a horizontal well equipped with ICDs producing oil from a homogeneous formation. A new equation with the form of a second order, non-linear Ordinary Differential Equation is derived to describe such a well. Our analysis of the equation provides both a precise numerical solution and an asymptotic solution. Practical engineering requires a compromise between the severity of the heel-toe effect and the reduction in the overall well performance. Hence, we have proposed a dimensionless criterion for estimating the optimal ICD type. Its application is illustrated by a real oil field-based example. Our new model is one of a few attempts to describe the performance and pressure distribution of an advanced (ICD equipped) well. It brings physical understanding of the well’s performance instead of treating it as a black box simulator. The asymptotic analytical solution gives a set of simple equations for the optimal ICD design of horizontal wells with strong heel-toe effects which can be implemented as a simple design sheet for the well completion engineer. The mathematical approach can be further extended to wells with advanced completion, such as those equipped with the groups of inflow devices or Interval Control Valves.
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Decoupled Overlapping Grids for Numerical Modelling of Transient Wellbore Pressure
Authors N. Ogbonna and D.B. DuncanAccurate modelling of transient wellbore pressure is important in well test analysis, where measured pressure responses are interpreted by comparing them to model results. Traditionally analytical models are used, but these models are limited to homogeneous reservoirs of regular shapes. Alternatively the wellbore pressure can be computed from a numerical simulation, allowing for more complex reservoir geometry and permeability heterogeneity to be included. The conventional point source or line source well approximation implemented in most commercial reservoir simulators does not provide good information about transient behaviour at the well. Another approach is to implement the well as an internal boundary with local refinement in the well vicinity. However this would significantly increase the computational cost of the iterative parameter fitting process in well testing, especially for field-scale models with large number of wells. This paper explores a new method for accurately computing wellbore and near-wellbore pressure. The method addresses the problem in two stages solved on different grids that overlap. In the first stage a global problem is solved in the entire domain with the conventional point/line source well approximation, and in the second stage a local problem is solved in a smaller near-well region with the well implemented as an internal boundary. The two solutions are linked via the external boundary condition for the local problem which is interpolated from the global solution. This method has the advantage of capturing both global reservoir properties, which can be accurately modelled using existing reservoir simulators, and the details of pressure transient phenomena associated with near-well refinement. The proposed method is validated against exact analytic solutions for a homogeneous 2D case study, and numerical results for some heterogeneous case studies are presented.
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A Genuinely One-dimensional Upwind Scheme with Accuracy Enhancement for Multidimensional Advection Problems
Authors A. Michel, Q.H. Tran and G. FavennecWe propose a nonlinear finite-volume scheme for the numerical approximation of the linear advection equation on multidimensional arbitrary grids. This scheme inspired from the two-step reconstruction philosophy of the anti-dissipative VoFiRe proposed by Despres, Labourasse and Lagoutiere but relies on a different procedure in the transversal step and a more propriate min-max principle in the longitudinal step. As a consequence, not only the stability analysis is simpler, but also a better accuracy is achieved on numerical results for any type of initial data. To illustrate the behaviour of the method we show some results concerning transport of tracers and two phase flow in porous media.
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Discretisation on Complex and Polyhedral Grids – Open Source MATLAB Implementation
Authors K.A. Lie, S. Krogstad, I.S. Ligaarden, J.R. Natvig, H.M. Nilsen and B. SkaflestadAccurate geological modelling of features such as faults, fractures or erosion requires grids that are flexible with respect to geometry. Such grids generally contain polyhedral cells and complex grid cell connectivities. The grid representation for polyhedral grids in turn affects the efficient implementation of numerical methods for sub-surface flow simulations. Methods based on two-point flux approximations are known not to converge on grids which are not $K$ orthogonal. In recent years, there has been significant research into mixed, multipoint, and mimetic discretisation methods that are all consistent and convergent. Furthermore, so-called multiscale methods have lately received a lot of attention. In this paper we consider a MATLAB implementation of consistent and convergent methods on unstructured, polyhedral grids. The main emphasis is put on flexibility and efficiency with respect to different grid formats, and in particular hierarchical grids used in multiscale methods. Moreover, we discuss how generic implementations of various popular methods for pressure and transport ease the study and development of advanced techniques such as multiscale methods, flow-based gridding, adjoint sensitivity analysis, and applications such as optimal control or well placement.
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Improving Stochastic Inversion Methods in History Matching Using Proxy Models
Authors K.D. Stephen and S. ArwiniAssisted history methods have been developed to automatically search the parameter space to find optimal models. Stochastic approaches increase the breadth of the search but can be very costly. Proxy models improve convergence by producing parameter sensitivities. Better distributions may be used in stochastic generation of new models. In this paper we present modifications to the neighbourhood and a genetic algorithm. Quadratic proxy models are derived for the misfit surface so that the search process can be made more efficient. We test the approaches on several analytical test functions. We also apply them to the Schiehallion field from the west of Shetland in seismic history matching. A quadratic regression equation was derived from a representative sample of models and used to generate gradients of the misfit with respect to the parameters. These are used to direct the choice of new models during a random search increasing the chance of finding new models. The proxy based method improves convergence by a factor of three generally. Complex surfaces see a lesser improvement though and the regression equation can be updated as better models are found to maintain the benefits.
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Explicit Singly Diagonally Implicit Runge-Kutta Methods and Adaptive Stepsize Control for Reservoir Simulation
Authors C. Völcker, J.B. Jørgensen, P.G. Thomsen and E.H. StenbySimulation of fluid flow in petroleum reservoirs is an essential tool in understanding, predicting and controlling advanced oil recovery methods. The major computational effort in reservoir simulation comes from solving a very large system of differential equations describing the fluid flow and the complex behaviour of advanced oil recovery methods. Choosing an appropriate method in the numerical solution of a large system of differential equations involves deciding on factors such as the order of the integration scheme, stability properties and concern for computational efficiency. Current simulators normally uses first order integration schemes applied with heuristically guided strategies for controlling time-step sizes. In the solution process of complex recovery methods this can lead to unnecessary computations and inappropriately small time-steps. We establish a fully implicit integrator of high order applied with an adaptive time-step selection supported by error estimates. We describe the explicit singly diagonally implicit Runge-Kutta (ESDIRK) methods with an embedded scheme for error estimation. This class of methods is computationally efficient, A- and L-stable as well as stiffly accurate. The embedded method providing the error estimate is of different order than the method used for advancing the solution. Based on this error estimate, the time-step is computed by a predictive control law. The predicitive control law is designed based on a model of the numerical method (ESDIRK) itself. Implicit integration involves the solution of a system of coupled nonlinear residual equations which need to be solved iteratively. Fast convergence of the iterative solver is crucial and may be controlled by the time-step size. We present a strategy for adaptive stepsize selection that mitigates the trade off between the convergence rate and time-step size. Consequently, the stepsize selection rule keeps the error estimate bounded (i.e., close to a user-specified tolerance) and at the same time maintains a good convergence rate of the equation solver. In addition, the controller has the ability to combine the above stepsize selection rule with classical time-step control that limits maximum change in key variables.
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An Application of Green’s Function Technique for Well Testing Horizontal and Partially Penetrating Wells
Authors D.V. Posvyanskii, E.S. Makarova, S.V. Milytin and V.S. PosvyanskiiThe purpose of well test analysis is to estimate reservoir properties from the measurement of wellbore pressure and production rate over time. Generally this is achieved by solving an inverse problem, the mathematical model of the reservoir generates the pressure response to compare with the actual one. The efficiency of solving the inverse problem depends both on the regression scheme and the number of evaluations of a direct task. The direct task is based on the solution of the diffusivity equation, which describes fluid flow in porous media. The paper focuses on the solution of the direct problem only. The diffusivity equation can be solved through the Green’s function (GF) method, where the solution is presented as a series over eigen values of the Laplace differential operator. However these series converge conditionally and the summation is time consuming which restricts the application of this technique for inverse problems in well test analysis. The same problems are well-known in quantum theory for solid state systems and the algorithm for fast summation of such series was proposed by Ewald. In [1] Ewald’s algorithm was successfully applied to well testing of vertical wells. In the present work we apply it to well test analysis of horizontal and partially penetrating wells. The application of Ewald’s algorithm allows for a fast and highly accurate solution. We have successfully applied this procedure for interpretation of real pressure buildup data taking account of sandface flow. Unfortunately, GF can be written out analytically only for simple cases. However using time measurements of pressure drop and production rate the Green’s function can be restored. This deconvolution technique is well known in well test analysis. In this study the reservoir response function was represented as decomposition on a basis of Daubechies orthogonal wavelets. This basis reduces the number of terms in the decomposition and so avoids problems connected with the regularization procedure [2]. For simple cases, the restored Green’s function is in a good agreement with the same analytical expressions. [1] E.S. Makarova, D.V.Posvyanskii, V.S.Posvyanskii, A.B. Starostin ECMOR XI P26 2008 [2] T.Schoroeter, F.Hollaender, A.C.Gringarten SPE 71574 2001
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Numerical Aspects of Using Vertical Equilibrium Models for Simulating CO2 Sequestration
Authors I. Ligaarden and H.M. NilsenStorage of CO2 in deep saline aquifers is considered an important means to reduce anthropogenic CO2 in the atmosphere. Assessing the risk of storage operations requires accurate modeling of migration of injected CO2. However, since potential injection sites typically are very large and time-scales long, flow simulation with traditional methods from the petroleum industry is often not feasible. Also, CO2 is very mobile and the flow is usually confined to thin layers, which put severe requirements on the vertical grid resolution. Using a vertical equilibrium assumption, the flow of a layer of CO2 can be approximated in terms of its thickness to obtain a 2D simulation model. Although this approach reduces the dimension of the model, important information of the heterogeneities in the underlying 3D medium is preserved. In this paper, we consider the Johansen formation, a candidate for CO2 sequestration, to compare the use of 3D simulations to simulations with a vertical equilibrium 2D model. We discuss numerical aspects of using the different methods, and demonstrate that the vertical equilibrium model provides more accurate results when the vertical grid resolution is low. Moreover, we investigate how averaging of parameters influences the accuracy of the vertically equilibrium solution.
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Impact of Geological Heterogeneity on Early-stage CO2 Plume Migration
Authors M. Ashraf, K.A. Lie, H.M. Nilsen and A. SkorstadInjection of CO2 into saline aquifers can be divided in two phases: injection and plume migration over long time scales. Large-scale injection operations should be preceded by simulation studies to determine how to maximize the injection volume/rate and minimize the leakage risk. Simulation of CO2 storage differs from oil recovery prediction not only in the objectives of study, but also in the characteristic length and time scales of the process. Working with long temporal and spatial scales and large uncertainties poses the question of how detailed the geological description should be. The impact of geological uncertainty on the oil production forecast has been thoroughly investigated in the SAIGUP study, where an extensive set of synthetic but realistic models of shallow-marine reservoirs were made. Tens of thousands of simulations were run for different production scenarios, demonstrating that variations in the structural and sedimentological description had a strong impact on production responses. Herein, we use the structural and sedimentological models from SAIGUP to study two questions related to CO2 storage: • How sensitive is the injection and early-stage migration to uncertainty and variability in the geological description? • What simplifying assumptions are allowed in averaging the geological attributes over scales?
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On Mathematical Simulation of Heavy Oil Recovery Problem
Authors A.K. Pergament and D.A. MitrushkinThe paper is dedicated to reservoir simulation of heavy oil and bitumen recovery using injection of heat-transfer agents (e.g. hot water or steam). Non-isothermal three-phase flow model, including oil, water, steam is considered. Difference scheme based on the finite volume method is proposed. The main feature of this problem is significant dependence of fluids properties on the temperature: the crucial point is a considerable decrease of oil viscosity with the increase of temperature. Consequently, it is necessary to accurately determine the temperature field close to the heat sources. The important result is that self-similar solutions of problems of steam injection and stationary gravity flow of three-phase fluid are constructed. The results of testing the derived difference scheme are represented for obtained self-similar solutions. The model problem of the heavy oil field development using the SAGD technology is discussed. The results of comparison of numerical simulation of temperature fronts on different computational grids are presented. We gratefully acknowledge the financial support of the Russian Foundation for Basic Research (grant 09-01-00823).
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Can Formation Relative Permeabilities Rule Out a Foam EOR Process?
Authors E. Ashoori and W.R. RossenFoam is a promising means of increasing sweep in miscible- and immiscible-gas enhanced oil recovery. SAG (surfactant-alternating-gas) is a preferred method of injection. Numerous studies verify that the water relative-permeability function krw(Sw) is unaffected by foam. This paper shows a connection between the krw(Sw) function and SAG foam effectiveness that is independent of the details of how foam reduces gas mobility. The success of SAG depends on total mobility at a point of tangency to the fractional-flow curve, which defines the shock front at the leading edge of the foam bank. Geometric constraints limit the region in the fractional-flow diagram in which this point of tangency can occur. For a given krw(Sw) function, this limits the mobility reduction achievable for any possible SAG process. The implications of this work include the following: Increasing nonlinearity of the krw function is advantageous for SAG processes, regardless of how foam reduces gas mobility. SAG is inappropriate for naturally fractured reservoirs if straight-line relative permeabilities apply, even if extremely strong foam can be stabilized in fractures. It is important to measure krw(Sw) separately for any formation for which a SAG process is envisioned.
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Improved Near-well Approximation for Prediction of the Gas/Oil Production Ratio from Oil-rim Reservoirs
Authors S.A. Halvorsen, A. Mjaavatten and R. AasheimStatoil has developed a model with short computational time to predict the rate dependent gas/oil ratio (GOR) from a horizontal well. The oil flow towards the wellbore is based on a one-dimensional model by Konieczek. The model performs remarkably well for medium time production optimization (weeks, months), while the predictions during the first days after a large change in the production can be poor. An improved one-dimensional model for the flow towards the wellbore is proposed, where the oil flow is treated as a superposition of three terms: 1) Radial flow towards the wellbore and towards a mirror well. 2) Flow to correct for modified boundary conditions due to the radial flows. 3) Flow due to height variations of the gas/oil contact (GOC). The new model takes care of the current short term and near-well deficiencies: Effect of 2D flow close to the wellbore, gas breakthrough due to viscous gas fingering, and horizontal/vertical anisotropy. Based on analysis and preliminary testing the new model should have equally good medium and long term capabilities and considerably improved short term and near-well behaviour, compared to the present implementation.
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Flow-based Grid Coarsening for Transport Simulations
Authors V.L. Hauge, K.A. Lie and J.R. NatvigGeological models are becoming increasingly large and detailed to account for heterogeneous structures on different spatial scales. To obtain computationally tractable simulation models, it is common to remove spatial detail by upscaling. Pressure and transport equations are different in nature and generally require different strategies for optimal upgridding. To optimize accuracy of a transport calculation, the coarsened grid should be constructed based on a posteriori error estimates and adapt to the flow patterns predicted by the pressure equation. Sharp and rigorous estimates are generally hard to obtain; herein we consider various adhoc methods for generating flow-adapted grids. Common for all, is that they start by solving a single-phase flow problem once and continue by agglomerating cells from an underlying fine-scale grid. We present several variations of the original method. First, we discuss how to include a priori information in the coarsening process, e.g., to adapt to special geological features or to obtain less irregular grids where flow-adaption is not crucial. Second, we show how different algorithmic choices can simplify the matrix structure of the discretized system and lead to reduced computational complexity. Finally, we demonstrate how to improve simulation accuracy by dynamically adding local resolution near strong saturation fronts.
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Multiscale Method for Numerical Simulation of Multiphase Flows in Giant Production Fields
Authors A.K. Pergament, V.A. Semiletov and P.Y. TominThe problem of multiphase flow modeling for giant oil and gas fields partitioned into several areas is considered. The aggregation of essential number of input fine grid cells forms the cell of coarse grid. According to ideas of I.Babuska, one can show that the pore pressure at each cell of the coarse grid may be approximated using linear combination of special basis functions. These functions are solutions of single-phase flow problem in the cell of the coarse grid with special piecewise multilinear functions used as boundary condition. The support operator method (SOM) by A.A.Samarsky is used to calculate the basis functions. Using R.P.Fedorenko idea of the superelement method (SEM), the calculated basis functions are used for approximation in SOM instead of ordinary linear function. Compared to SEM, the number of basis functions for method developed is substantially smaller: not 8 but 3 for the hexahedron grid. Finally the distribution of pressure and saturation evolution is calculated with Neumann boundary conditions for governing system. Method developed is high-resolution one and allows effective simulation of the processes for giant production fields. We gratefully acknowledge the financial support of the RFBR (grant 09-01-00823).
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Multiscale Wavelet Analysis Applied to Localization of Covariance Estimates in EnKF
Authors O. Pajonk, R. Schulze-Riegert, M. Krosche and H.G. MatthiesThe ensemble Kalman filter (EnKF) has become very popular in the field of assisted history matching for its appealing features. Nonetheless, problems can result from so-called spurious correlations due to the finite ensemble size [e.g. Evensen, 2009], which are considered as unphysical. The result of these correlations is a reduction of ensemble spread at model locations where no related data is available. This may cause an underestimation of the uncertainty and can result in a collapsed ensemble [Hamill, 2001]. Two methods are commonly used to address the unwanted reduction of variance: covariance infla-tion and localization. This contribution presents a new covariance localization approach based on multiscale (or multiresolution) wavelet analysis [Daubechiers, 1992]: the model state vector is transformed to a multiscale wavelet space. Correlations are computed in this space, not in the model space. This procedure allows the application of a new localization scheme, i.e., a different covariance localization function can be applied for each of the scale levels using a standard Schur product approach. Especially it allows us to filter unphysical long range correlations from fine scales while retaining longer correlations on coarser scales. Afterwards EnKF updates are computed and the transformation back to model space is applied. This contribution explains our wavelet-based localization approach and presents numerical results for the application of a synthetic model. The results are compared to standard localization approaches. The application to a real field simulation model is discussed.
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