A spatial-temporal approach to reduced complexity modelling for hydrocarbon reservoir optimization Insuasty Moreno, E.G.

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1 A spatial-temporal approach to reduced complexity modelling for hydrocarbon reservoir optimization Insuasty Moreno, E.G. Published: 31/05/2018 Document Version Publisher s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: A submitted manuscript is the author's version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. The final author version and the galley proof are versions of the publication after peer review. The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal? Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Download date: 17. Jul. 2018

2 A SPATIAL-TEMPORAL APPROACH TO REDUCED COMPLEXITY MODELLING FOR HYDROCARBON RESERVOIR OPTIMIZATION PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de Rector Magnificus prof.dr.ir. F.P.T. Baaijens, voor een commissie aangewezen door het College voor Promoties, in het openbaar te verdedigen op donderdag 31 mei 2018 om 11:00 uur door Edwin Giovanni Insuasty Moreno geboren te San Juan de Pasto, Colombia

3 Dit proefschrift is goedgekeurd door de promotoren en de samenstelling van de promotiecommisie is als volgt: voorzitter: Prof.dr.ir. Bart Smolders 1 e promotor: Prof. dr. ir. Paul M.J. Van den Hof 2 e promotor: Prof. dr. ir. Jan-Dirk Jansen (Technische Universiteit Delft) copromotor: Prof. dr. Siep Weiland leden: Prof. dr. ir. Arnold W. Heemink (Technische Universiteit Delft) Prof. dr. William R. Rossen (Technische Universiteit Delft) Prof. dr. Wil H.A. Schilders adviseur: dr. Tzu-hao Yeh (Shell Global Solutions, Houston, USA) Het onderzoek of ontwerp dat in dit proefschrift wordt beschreven is uitgevoerd in overeenstemming met de TU/e Gedragscode Wetenschapsbeoefening.

4 A spatial-temporal approach to reduced complexity modelling for hydrocarbon reservoir optimization Edwin Giovanni Insuasty Moreno

5 The research reported in this thesis is part of the research program of the Dutch Institute of Systems and Control (DISC). The author has successfully completed the educational program of the Graduate School DISC. The work presented in this thesis has been supported by the Recovery Factory program, funded by Shell Global Solutions International. A spatial-temporal approach to reduced complexity modelling for hydrocarbon reservoir optimization by Edwin Giovanni Insuasty Moreno. A catalogue record is available from the Eindhoven University of Technology Library. ISBN: This thesis was prepared with the L A TEX documentation system. Reproduction: Gildeprint, Enschede, The Netherlands. Copyright c 2018 by E.G. Insuasty Moreno. All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

6 Summary A spatial-temporal approach to reduced complexity modelling for hydrocarbon reservoir optimization Simulation of multiphase flow through porous media and reservoir models are central for the modern practice of reservoir engineering. Despite of the advances in computing power and of the advent of smart field technologies, the implementation of model-based operational/control strategies for oil production, and the development of methods for parameter estimation of petrophysical properties have been limited by the inherent complexity of reservoir models. This thesis develops methods to generate control-relevant low-complexity reservoir models, in order to accelerate the optimization and parameter estimation loops for closed-loop reservoir management. In reservoir engineering, it is attractive to characterize the difference between reservoir models in metrics that relate to the economic performance of the reservoir as well as to the underlying geological structure. The first part of this thesis elaborates on the development of a dissimilarity measure that is based on reservoir flow patterns under a particular operational strategy. To this end, a spatial-temporal tensor representation of the reservoir flow patterns is used, while retaining the spatial structure of the flow variables. This allows reduced-order tensor representations of the dominating patterns and simple computation of a flow-induced dissimilarity measure between models. The developed tensor techniques are applied to cluster model realizations in an ensemble, based on similarity of flow characteristics, to define reduced-size ensembles, and to use them to accelerate model-based operations (robust production optimization and data assimilation). In this work, the concept of model complexity has been assessed from the dynamic and static perspectives. On the one hand, the concept of dynamical complexity is understood in the context of simulation of large scale dynamical systems generated after the spatial discretization of the governing partial differential equation (PDEs), and tensor decompositions and representations of flow profiles are used to characterize empirical features of flow simulations. The concept of classical Galerkin projection is extended to perform projections of flow equations onto empirical tensor subspaces, generating in this way, reduced order approximations of the original mass and momentum conservation equations. v

7 vi Summary On the other hand, the static/geological complexity is understood in the context of parameterization of petrophysical properties, and tensor methods are used to represent and approximate petrophysical properties. The efficiency of tensor approximations when representing channelized structures is verified and compared with classical techniques such as Principal Component Analysis (PCA) parameterizations. Parameter estimation experiments involve the estimation of permeability fields of a channelized reservoir using three alternative parameterizations: grid block, lowdimensional PCA and tensor-based, while the uncertainty space is covered by an ensemble of channelized realizations. When using tensor representations, the geological realism of the updated ensemble is kept while having prediction capabilities similar to the one achieved with the PCA approach.

8 List of Abbreviations PDE ODE S&C NPV CLRM POD CAHM SVD PCA I/O TOF FV FE USD AD MRST AD-GPRS RO KF EnKF CPD TSVD HOSVD MSVM Partial Differential Equation Ordinary Differential Equation Systems and Control Net Present Value Closed Loop Reservoir Management Proper Orthogonal Decomposition Computer-Assisted History Matching Singular Value Decomposition Principal Component Analysis Input/Output Time-Of-Flight Finite Volume Finite Element United States Dollars Automatic Differentiation Matlab Reservoir Simulation Toolbox Automatic Differentiation General Purpose Reservoir Simulator Robust Optimization Kalman Filter Ensemble Kalman Filter Canonical Polyadic Decomposition Tensor Singular Value Decomposition High Order Singular Value Decomposition Maximum Singular Value Modal Rank vii

9 viii List of Abbreviations SDM MDS stb STOIIP CDF Single Directional Modal rank Multidimensional Scaling stock barrel Stock Tank Oil Initially in Place Cumulative Density Function

10 List of Figures 1.1 Predictions for the demand of primary energy, adapted from the Shell Report [2008] Diagram of the oil production for the primary and secondary recovery stages Schematic of the water-flooding process using horizontal injection and production wells. Adapted from Brouwer and Jansen [2004] Diagram for the CLRM approach, adapted from Jansen et al. [2009a] Classical reduced order modeling procedure Rock permeability and well configuration of 6 realizations of an oil reservoir, adapted from Jansen et al. [2014] Rock permeability and well configuration of two reservoir models with very similar dynamical response (solid and dashed lines), adapted from van Essen et al. [2016] Relative permeabilities Schematic description for the truncation of a Tucker decomposition of a 3D tensor Time snapshots for oil saturation and (10, 10, 10) approximation computed using TSVD, HOSVD, MSVM and SDM Distribution of the computational time for tensor and matrix decomposition algorithms. HOSVD of S and SVD of S were repeated 1000 times. Green: HOSVD. Blue: SVD Computational time for the matrix and tensor reconstruction methods. Reconstruction of S and of S from decompositions was repeated 1000 times. Green: HOSVD. Blue: SVD Schematic of the tensor representation of a reservoir flow pattern. Axes represent spatial-temporal coordinates. Color-scale corresponds to oil saturation ix

11 x List of Abbreviations 3.2 Oil-water front with tensor approximations. Approximation 1 has modal rank (20, 20, 5). Approximation 2 has modal rank (10, 10, 5). Colors represent oil saturation Blue-Left axis: Relative proximity ν(r). Green-Right axis: Size in memory of the modal rank approximation (r, r, 5) as function of r Schematic interpretation of a 4D tensor of reservoir flow patterns Well configuration and samples of permeability fields from the ensemble. Color scale in mdarcy Approximation error e r (Î, Ĵ) = S Ŝ(Î,Ĵ) F S F Snapshots of oil saturation for model 57 (layer 3) with Î = 10, Ĵ = 10, Ẑ = 2, ˆK = 2 and ˆR = 100. Top: Reservoir simulation. Bottom: Tensor approximation MDS plot. Color represent flow-based clusters. Numbers are assigned to all the realizations MDS plot. Left: Color represents NPV (USD). Right: Color represents total oil production (stb). Numbers are assigned to all the realizations Snapshots of oil saturation (top layer) of sample models from Cluster c Snapshots of oil saturation (top layer) of sample models from Cluster c Snapshots of oil saturation (top layer) of sample models from Cluster c Snapshots of oil saturation (top layer) of sample models from Cluster c Model clusters similarity with respect to clusters of the base case Flow-relevant ensemble Geometry and well configuration of the Brugge field MDS plot. Color represent flow-based clusters. Numbers are assigned to all the realizations. Left: 5D tensor. Right: 6D tensor Snapshot of oil saturation front (top layer) of sample models from different clusters after 30 years of production. Background colors represent clusters Snapshot of oil saturation front (top layer) of sample models from different clusters after 30 years of production. Background colors represent clusters CDF of total oil production using 15 realizations. For every flow measure, the CFD is computed 1000 times. Surfaces show the dispersion around the real CFD (black line). Blue: Random sampling. Yellow: TOF. Red: 5D Tensor. Green: 6D Tensor. Black line: full ensemble Variance of cum oil distribution for all the percentiles. 15 realizations. 77

12 List of Abbreviations xi 4.8 CDF of total oil production using 50 realizations. For every flow measure, the CFD is computed 1000 times. Surfaces show the dispersion around the real CFD (black line). Blue: Random sampling. Yellow: TOF. Red: 5D Tensor. Green: 6D Tensor. Black line: full ensemble Variance of cum oil distribution for all the percentiles. 50 realizations CDF of total oil production using 85 realizations. For every flow measure, the CFD is computed 1000 times. Surfaces show the dispersion around the real CFD (black line). Blue: Random sampling. Yellow: TOF. Red: 5D Tensor. Green: 6D Tensor. Black line: full ensemble Variance of cum oil distribution for all the percentiles. 85 realizations History matching and rate predictions. Total rates at well 21. Green background shows history matching interval. Pink background shows prediction interval History matching and rate predictions. Total rates at well 23. Green background shows history matching interval. Pink background shows prediction interval History matching and rate predictions. Total rates at well 24. Green background shows history matching interval. Pink background shows prediction interval History matching and rate predictions. Total rates at well 28. Green background shows history matching interval. Pink background shows prediction interval Left: Permeability (md) and injectors/producers location for one realization. Right: Samples of the ensemble MDS for the low dimensional flow profiles from the ensemble. Colors represent final NPV. Left: 3D map. Right: 2D map Saturation patterns and NPV build-up for realizations with highly dissimilar flow Saturation patterns and NPV build-up for realizations with similar flow Injection and production rates for RO. Blue/Red: Full ensemble (R = 1000). Black: Strategy with control-relevant ensemble (R red = 50) Distribution of NPV over the ensembles Left: Optimal production strategy for Reservoir model 1. Right: Optimal production strategy for Reservoir model 2. Blue: Injector rates 1. Green: Injector rates 2. Red: Producer rate 1. Cyan: Producer rate MDS for Nominal Optimization. Left: Optimal production strategy for Reservoir model 1 applied to the ensemble. Right: Optimal production strategy for Reservoir model 2 applied to the ensemble. Color scale represent final NPV

13 xii List of Abbreviations 5.1 Permeability, porosity for the test case POD and Tensor basis for saturation Oil saturation time snapshots for full model and reduced order model Optimal production strategies for full order and reduced order approximations NPV performance of the virtual asset with the different strategies Rock permeability and well configuration of two updated reservoir models with very similar dynamical response (solid and dashed lines), adapted from van Essen et al. [2016] Schematic of the tensor representation of a petrophysical property (permeability) of an ensemble of channelized models. The horizontal axes represent spatial coordinates and the vertical axis the model coordinate Low rank approximations using classical SVD and tensor decompositions Relative error of the approximations. Left: Relative error as a function of the basis functions in the x and y coordinates. The color-scale corresponds to magnitude. Right: Relative error fixing the number of basis functions in one coordinate, and varying the number of basis functions in the other A realization of the permeability field and well configuration of the reservoir models. Red: Producer wells. Black: Injector wells History matching and rate prediction. Total rates at well 5. The green background indicates the history matching interval. The pink background indicates the prediction interval. Red: Truth. Gray: Updated ensemble History matching and rate prediction. Total rates at well 5. The green background indicates the history matching interval. The pink background indicates the prediction interval. Red: Truth. Gray: Updated ensemble History matching and rate prediction. Total rates at well 11. The green background indicates the history matching interval. The pink background indicates the prediction interval. Red: Truth. Gray: Updated ensemble History matching and rate prediction. Total rates at well 15. The green background indicates the history matching interval. The pink background indicates the prediction interval. Red: Truth. Gray: Updated ensemble Final update permeability field, realization

14 List of Abbreviations xiii 6.11 Final update permeability field, realization History matching and water breakthrough prediction. Fractional flow at well 5. The green background indicates the history matching interval. The pink background indicates the prediction interval. Red: Truth. Gray: Updated ensemble History matching and water breakthrough prediction. Fractional flow at well 9. The green background indicates the history matching interval. The pink background indicates the prediction interval. Red: Truth. Gray: Updated ensemble

15 xiv List of Abbreviations

16 List of Tables 2.1 Relative approximation error of various algorithms for tensor decomposition Cluster distribution of NPV and total oil Production strategies for the sensitivity analysis Characteristics of the flow patterns Details of the experimental setting Rock and fluid properties for the model in Section Physical parameters Details of the experimental setting xv

17 xvi List of Tables

18 Contents Summary List of Abbreviations v vii 1 Introduction The energy landscape Oil production Reservoir models and simulation A model-based approach to oil production Challenges to the model-based approach to oil production Control-relevant reservoir models Discussion Research objective and directions Thesis outline Reservoir modeling, optimization and algebraic techniques Introduction Reservoir models State space representations Reservoir simulator Optimization methods for model-based operations of oil reservoirs Introduction The adjoint equations and gradients Production Optimization Parameter estimation Tensor modeling and decompositions Fundamentals of linear and multilinear algebra xvii

19 xviii Contents Tensor decompositions and approximation Algorithms for tensor decompositions Computational complexity of tensor decompositions Concluding remarks I A spatial-temporal approach for flow characterization 39 3 Tensor formulation of flow-based dissimilarity measures Introduction Flow-based dissimilarity measures Introduction Dissimilarity measures and distance functions Low dimensional representations and flow-based distances through SVD Discussion Spatial-temporal tensor methods for flow-based dissimilarity measures Introduction Tensor representations of reservoir flow patterns Tensor approximation of reservoir flow patterns D Tensors: an approach for handling multiple realizations Flow-based dissimilarity measures in low-dimensional tensor representations Discussion A tensor-based workflow for model clustering using flow measures Introduction k-means tensor clustering Visualization of dissimilarities A workflow for model clustering using flow measures Application case Generation of the reservoir flow patterns Low-dimensional tensor representation of the reservoir flow patterns Model clustering and visualization NPV and oil production in the clusters Input dependency of the reservoir flow patterns and dissimilarity measures Concluding remarks

20 Contents xix 4 Reduced-size ensembles in CLRM with flow-based dissimilarity measures The selection of representative realizations Assessing the quality of the reduced-size ensembles The Brugge field Flow-based model classification Assessing the quality of the flow-relevant ensembles Acceleration of the parameter estimation stage using reduced-size ensembles Introduction Parameter estimation using flow-relevant ensembles Discussion Production optimization with flow-relevant ensembles The reservoir model Generation of the flow-relevant ensembles Production optimization under geological uncertainty using flow-relevant ensembles Discussion The Effect of the Production Strategy on the Generation of Flow- Relevant Ensembles Concluding remarks II A tensor approach for the reduction of dynamical complexity 95 5 Tensor methods for the model reduction of reservoir models Introduction Classical empirical projection techniques for model order reduction Classical spectral expansions POD basis functions The Galerkin projection Tensor-based reduced order modeling Tensor representations and decompositions Nested Galerkin projections Tensor-based model order reduction of a single-phase reservoir model Application of the nested Galerkin projection Application case: Tensor-based model order reduction for multiphase flow models

21 xx Contents The model Snapshots generation and decomposition Model reduction Tensor-based reduced-order adjoint models for production optimization Production optimization Reduced order adjoint equations Results Concluding remarks III A tensor approach for the reduction of geological complexity Estimation of petrophysical parameters using tensor representations Tensor modeling and decompositions for the representation of petrophysical properties Tensor approximations of permeability fields Tensor parameterizations of permeability fields EnKF for parameter estimation using tensor parameterizations Application case Conclusion Contributions and future work Part I: Spatial-temporal approach for flow characterization Part II: A tensor approach for the reduction of dynamical complexity Part III: A tensor approach for the reduction of geological complexity Final remarks Bibliography 142 Acknowledgments 157 Curriculum Vitae 159

22 1 CHAPTER Introduction 1.1 The energy landscape The ability of managing energy resources has boosted the technological development of our contemporary civilization and has enabled humans to increase their life expectancy and population. The human development index of the United Nations, in the UN Report [2015] indicates the existence of a positive correlation between the energy consumption and quality of life in developed countries, which has pushed the energy industry to efficiently produce large and increasing amounts of energy in order to sustain the human development. In this context, hydrocarbon resources play a key role in the world s energy landscape nowadays, and their sustainable and responsible exploitation is subject of debate among governments, environmental organizations, communities and major oil companies around the world. In Fig. 1.1, the quantitative projections for the demand of oil and gas shows an increasing trend until 2020, and a slight decay until 2050 with the advent of renewable energy technologies, see the Shell Report [2008]. Despite this scenario, fossil fuels constitute the main source of energy for the next decades, peaking around 80% and reaching almost 60% of the total demand in 2020 and 2050 respectively. Considering oil and gas only, the combined demand remains steady in the next decades, see Fig Keeping a steady oil and gas production constitutes a big challenge for the international oil companies (e.g. Shell, Exxon, etc.), as they are facing the end of the cheap-to-produce oil: conventional fields which are geographical and geological accessible, and close to refining facilities and consumers, are nearly depleted, and the exploration of new oil and gas fields at accessible locations are becoming more scarce. These facts have encouraged the use of more efficient recovery strategies for the new and existing oil fields. 1

23 2 Introduction Figure 1.1: Predictions for the demand of primary energy, adapted from the Shell Report [2008]. 1.2 Oil production The target of oil production is the exploitation of a hydrocarbon resource in an economic way. Petroleum is produced from reservoirs beneath the surface of the earth and it resides in porous rock formations, which are found at the exploration stage. Part of the exploration consists of analyzing geological cores, and the analysis of the arrival time and frequencies of the acoustic waves using geophones and seismic technology to estimate the geological structure of the reservoir, see, e.g., Russell and Dommico [1988]. With this information, the problem of identifying the geological structures of the subsurface translates into an inverse modeling and signal processing problem, where measured input (seismic signals) and outputs (reflected wave signals on different measurement locations) are used to estimate a geological formation which explains the patterns in the data. This is an ill-posed problem, as the number of parameters largely exceeds the available measurements and there might be several solutions to it. Usually, geophysicists and reservoir engineers provide several geological interpretations that match the measured data. With this analysis, exploratory wells are drilled in order to assess whether the recovery is technically and economically feasible. The production stage usually consists of several phases. In the primary stage, the potential energy of the reservoir is exploited to produce the oil, which naturally flows

24 1.2 Oil production 3 to the surface due to the pressure gradient. Currently, roughly 5% of the oil and gas production wells in the world are operated in this way. After some years of operation, the internal pressure of the reservoir decays, leading to a proportional decrease of the oil production, as it is depicted in Fig Figure 1.2: Diagram of the oil production for the primary and secondary recovery stages. Low oil production motivates the use of techniques for artificial lift and water injection in a secondary stage. In one particular secondary recovery method, called water-flooding, the reservoir is flooded with water to increase the potential energy inside the reservoir and to displace the oil towards the production wells. This is a well established standard method for secondary recovery in the petroleum industry, allowing between 15% to 40% of the oil in place to be recovered. The secondary recovery is usually completed when the amount of water at the production wells is high enough to make the reservoir operation financially unfeasible. One of the biggest challenges in the water-flooding phase is to have an uniform and controlled front. Initially, water is injected homogeneously through the injection wells, but as the flow evolution depends on the heterogeneous petrophysical properties like permeability, the oil-water front may have a non-uniform shape, which often leads to flow conditions as the one depicted in Fig The reservoir geology is usually composed by streaks of high permeability where the fluids can easily flow, generating fluid transport in the form of fingers, which often reach the production wells faster than the bulk of remaining oil in place. The early water breakthrough due to the fingering phenomena at the producers influences the decisions taken by operators at surface facilities, which may consider shutting down wells with low oil/water production ratio, and loosing in this way production capacity by leaving the vast majority of the reserves volume in situ. This scenario may affect the strategic plans of the oil company and governmental agencies which usually try to maximize the oil recovery and net profits in the long term. The production strategy determines the future performance of the reservoir, and should consider the short-term (financial) and the long-term (strategic) objectives of the oil company. A classical approach for oil production is by making use of a reactive strategy. In this method, production wells are shut in if they are no longer financially feasible, i.e., the costs of water production surpasses the profits gener-

25 4 Introduction Figure 1.3: Schematic of the water-flooding process using horizontal injection and production wells. Adapted from Brouwer and Jansen [2004]. ated by the produced oil. This method has proven to be very effective in achieving short-term financial goals, however, the oil sweep is usually poor, leaving out a large amount of oil in place. In the past decade, a model-based approach for operation of oil reservoirs has been developed, see, e.g., Jansen et al. [2008]. In the model-based approach for operation of oil reservoirs, it is possible to embed objectives such as the profits in the longterm, while using the existing reservoir models to forecast the future behavior of the reservoir variables. This approach is described in the next sections. 1.3 Reservoir models and simulation Reservoir simulation plays a prominent role in the modern practice of reservoir engineering. Reservoir models describe the multiphase flow through porous media with possible thermal and geomechanical interactions. The transport phenomena equations for reservoir models consist of mass, momentum and energy balances. For multiphase flow models, the mass balance is described by a set of partial differential equations (PDEs) for every phase, see, e.g., Aziz and Settari [1979]. The (degenerated) momentum equation is described by Darcy s law, which contains only gravity and resistance terms, while neglecting inertial forces by assuming slow displacement of the oil-water front. The flow variables to be considered are usually phase saturations, pressure, temperature, etc. See Chapter 2 for a more detailed description of the physics considered in this work. The transport equations for multiphase flow through porous media have a continuous nature. To derive a reservoir simulator, the spatial domain of the PDEs is discretized using a finite grid, generating a large-scale system of ordinary differential equations ( equations) which describe the temporal evolution of the flow variables under consideration. There are several methods to derive simulation models for oil reservoirs. Traditionally, the Finite Volume method has been used as the standard technique by the oil industry, as the conservation laws of mass and momen-

26 1.4 A model-based approach to oil production 5 tum still apply to the grid-based balances, see, e.g., Aziz and Settari [1979]. However, Finite Element methods are becoming popular nowadays because of their ability to deal with unstructured grids and multi-scale phenomena, see, e.g., Chavent and Jaffre [1986]. The discrete form that results from the spatial and temporal discretization of the transport equations will serve as a basis for the model-based operations described in the next section. 1.4 A model-based approach to oil production The increasing demand for energy has encouraged the use of improved production strategies for conventional oil and gas resources. Recently, the advent of smart field technology has boosted the possibilities of actuation and sensing, and there is a significant scope for the oil companies to achieve long-term strategic objectives with the aid of these new developments, see, e.g., Foss [2012]. In the last decade, there has been a significant effort to increase the use of reservoir models for the management of reservoir assets, with the aid of Systems and Control (S&C) theory, see, e.g., Van Doren [2010], Kaleta et al. [2011] and Jansen [2013] among others. In this context, several studies have indicated a significant scope for reservoir model-based life-cycle optimization of ultimate recovery or Net Present Value (NPV), especially when combined with computer-assisted history matching leading to a closed-loop reservoir management (CLRM) approach; see, e.g., Jansen [2013],Jansen et al. [2009a], Sarma et al. [2008b] or Chen et al. [2009a]. In this CLRM approach the assessment of the value of information of different resources becomes an important feature, see e.g. Barros et al. [2015]. From the S&C perspective, oil fields can be categorized as batch processes, Forgione et al. [2015a]. Batch processes produce a fixed amount of output product for a fixed production time, using a sequence of control actions. On the one hand, the advanced control of batch processes starts with the definition of a performance-related cost function to be maximized. This step aids the generation of optimal trajectories for the physical variables and control inputs related to the process, while considering state and input constraints. On the other hand, the static parameters of the physical model are usually unknown, and have to be estimated during the operation using the information provided by sensors using a data-driven approach, see e.g. Forgione et al. [2015b]. The closed-loop reservoir management approach (CLRM), see, e.g., Jansen et al. [2009a], uses concepts from S&C theory to operate oil fields as batch processes, by including model-based control and parameter estimation loops as the key elements of the decision making process. A graphical description of the CLRM process is displayed in Fig On the one side, the control loop involves the definition of a long-term strategic objective, such as the maximization of the Net Present Value (NPV) or the total oil recovery. Here, reservoir models are used as constraints in the optimization problem, while considering rates and pressure constraints at the injector and producer wells. See Chapter 2 for details. The purpose of this stage is to generate an optimal production strategy (control settings) which maximizes a performance-related measure.

27 6 Introduction Figure 1.4: Diagram for the CLRM approach, adapted from Jansen et al. [2009a]. On the other side, the parameter estimation loop includes several types of measurements such as tubing head pressures, and water and oil rates at injectors and producers, and (occasionally) seismic data to estimate the location of the oil-water front. These methods are used to estimate the petrophysical properties of the subsurface such as permeability, porosity, net-to-gross ratio, etc. Reservoir models are used as constraints in an optimization problem that aims for the minimization of the difference between past output measurements and the outputs of the reservoir simulator. Another approach for parameter estimation uses a model-based filtering approach, see, e.g., Evensen [2009], where Kalman gains are designed to estimate states and parameters simultaneously, see Chapter 2 for technical details. 1.5 Challenges to the model-based approach to oil production Dynamical complexity Reservoir models are typically described by a set of nonlinear PDEs which require numerical methods to find feasible solutions. After numerical discretization, the dynamical systems contain thousands to millions of state variables, and can be classified as large-scale dynamical systems. Due to the computational cost of large-scale reservoir simulations, parameter estimation and model-based procedures for production optimization are limited by computational capabilities and time constraints. Hence, there is a clear need for reservoir models with reduced dynamical complexity

28 1.5 Challenges to the model-based approach to oil production 7 that can be used for efficient simulation of fluid flow through porous media. Model order reduction has been a prolific field in the last few decades, with applications in virtually every field of engineering where the dynamical complexity of mathematical models limits their application in model-based operation. Figure 1.5: Classical reduced order modeling procedure. The classical procedure for model order reduction starts with the discretized versions of the governing PDEs using numerical methods such as finite volumes, elements or differences. Subsequently, projection mechanisms are applied to derive reduced order models, see Fig. 1.5 for a schematic procedure and classical methods. While model reduction of linear systems has reached a certain level of maturity (see, e.g., Antoulas et al. [2001] for a complete survey for linear (Balanced, Hankel and Krylov methods) and nonlinear (POD method) models), the model reduction of nonlinear dynamical systems is still a relevant research topic in the mathematical community. In reservoir engineering, Proper Orthogonal Decomposition (POD) has been proven to be an efficient tool for model order reduction of large-scale nonlinear dynamical systems. POD techniques have been exploited by several authors, see, e.g.,heijn et al. [2004], Markovinovic and Jansen [2006], Krogstad [2011]. In production optimization, Van Doren et al. [2006] have used a projection onto POD spaces of the adjoint equations to reduce the dimension of the linear adjoint system.gildin et al. [2006] have designed optimal control strategies based on POD models. In parameter estimation, Kaleta et al. [2011] have used the adjoint of a reduced order linearization of a full order model to perform history matching procedures. To accelerate reduced

29 8 Introduction order models, Astrid [2004] has developed the Missing Point Estimation technique to select relevant equations for the projection step. For reservoir simulation, in the works of Cardoso et al. [2009, 2010a,b] the linearization of the reservoir equations along a trajectory has been computed and projected onto a reduced set of equations, to accelerate the reduced order model. Despite the benefits of reduced order models for reservoir simulation, production optimization and history matching, various authors have reported a recurrent limitation of the state-of-the-art methods for model order reduction: most of the projection techniques for realistic reservoir models generate, in most of the cases, either unstable or inaccurate reduced reservoir models, see the work of Heijn et al. [2004] and Cardoso et al. [2010a]. Geological uncertainty The subsurface is usually poorly understood, as the only precise and reliable source of knowledge of the petrophysical properties is obtained from cores or logs at exploration wells. Typically there are several sources of uncertainty in the geological modeling stage, see e.g. Caers [2011], particularly: 1. Interpretation of the seismic data: The data quality and the interpretation of seismic information is one of the major sources of uncertainty. The seismic information is usually translated into depth data using an uncertain wave velocity model, which defines fixed (deterministic) geological structures in the subsurface. This interpretation depends largely on the experience and skill of the geophysicists, moreover, the interpretation of seismic data may drastically change if a different analyst is considered, thus resulting in uncertainty in the geometry and structure of the field, see, e.g., Bond et al. [2007] and Polson and Curtis [2010]. 2. Measurement errors: In geological modeling, the measurements are considered as approximate estimates of the variables in consideration. In practice, the precision of sensor devices depends on their calibration, and the sensitivities bands which are provided by the manufacturer. In addition, there are measurement errors induced by the environmental conditions at the locations where the measurements were taken. Despite the fact of having incomplete knowledge of the petrophysical properties, it is possible to provide a quantification of the spatial uncertainty of the geology of oil fields. Geostatistics provides a set of methods to understand the spatial variability and connectivities of the rock formations at the subsurface, allowing geoscientists to contribute to the reservoir characterization process by defining realistic (but uncertain) realizations of the petrophysical properties. These properties constitute model parameters and are one of the most relevant parts of the reservoir simulator, as they largely determine the flow behavior in the oil field. Typically, classical methods for

30 1.5 Challenges to the model-based approach to oil production 9 geostatistical simulation allow the generation of many (thousands) realizations following a statistical approach, where the uncertainty corresponding to the parameter space is accounted by taking samples of the statistical distributions of the petrophysical properties. These equally probable samples are collected to create an ensemble of realizations, which tries to capture the geologically realistic features of the subsurface, see Fig. 1.6 as an example of reservoir realizations and the typical variability found in realistic fields. Figure 1.6: Rock permeability and well configuration of 6 realizations of an oil reservoir, adapted from Jansen et al. [2014]. To properly account for the effect of geological uncertainty, it is important to perform both the history matching, life-cycle optimization and value of information assessment on the basis of several realizations of the reservoir model. For instance, in the dynamical context, dealing with multiple (hundreds to thousands) of reservoir models, and use them for the control and parameter estimation loops of the CLRM scheme is technically impossible, and the selection of representative models has become a relevant issue for the practice of model-based operations in reservoir engineering. Static complexity One approach to reduce the level of uncertainty of the petrophysical properties consists of the application of data-driven methods for parameter estimation, a procedure called Computer-Assisted History Matching (AHM) in the petroleum community. CAHM aims to assimilate production data by finding an updated set of model parameters (permeabilities, porosities, net-to-gross ratios, etc.), such that the mismatch between production data and the model response is minimized. Typically, the number of parameters to be estimated exceeds by far the number of available field measurements, and the parameter estimation problem becomes ill-conditioned. As the

31 10 Introduction parameter estimation procedure is posed as an ill-conditioned optimization problem, there are usually several optimal solutions depending on the given initial condition, i.e., there are multiple realizations of the petrophysical properties that minimize the mismatch between the model output and the data. This implies that, while two history-matched reservoir realizations may look very different from the geological perspective, they may have very similar outputs in their response space, see Fig Figure 1.7: Rock permeability and well configuration of two reservoir models with very similar dynamical response (solid and dashed lines), adapted from van Essen et al. [2016]. From a Systems and Control perspective, a parameterized mathematical model that, when provided with different parameter sets, generates similar responses is usually classified as a non-identifiable model. Typically, reservoir models are non-identifiable, see e.g.van Doren et al. [2009, 2011]. In simple words, the petrophysical parameters are currently described at a level of detail which is too high to achieve a proper validation from the data. This problem is usually addressed through the use of a prior term with spatial correlations described with two-point statistics (a covariance matrix). However, spatial features that extend over many grid blocks, such as channels or faults, are not captured by such a description. In order to address this issue, several authors have developed parameterization methods of geological structures which include techniques based on sensitivity analysis and PCA: Oliver [1996],Reynolds et al. [1996],Tavakoli et al. [2010]; Wavelets: Sahni et al. [2005]; Channel parameterizations: Van Doren et al. [2011]; Discrete Cosine Transform: Jafarpour et al. [2009, 2010]; Pluri-Principal Component Analysis: Chen et al. [2016]. Recently, there have been several developments aimed at preserving geological realism after the CAHM step, see, e.g., Vo and Durlofsky [2014, 2015]. Sarma et al. [2008c] have introduced the kernel PCA method, which can capture, to some extent, higherorder statistics, but involves the solution of a nontrival pre-imaging problem. From the perspective of CAHM, parameterizations are attractive because: 1. Identifiability: They increase the identifiability of the inverse modeling problem by estimating fewer parameters. 2. Spatial structure: They sometimes maintain spatial features such as channels or faults, that are not captured by two-point statistics (covariances).

32 1.6 Control-relevant reservoir models 11 Despite of these efforts, finding natural and simple representations of the geological structures is still an active field of research in the reservoir engineering community. We have already defined the notions of dynamical and static complexities. In this dissertation, the scope of reservoir complexity will be limited to the notions discussed in this section. In the next section, we elaborate on the problem of reducedcomplexity reservoir models, which are control-relevant for CLRM. 1.6 Control-relevant reservoir models Previously, we have motivated the generation of reservoir models with reduced complexity, and we have cited several studies that indicate that the model-based management of oil reservoirs can be performed using simple models. This fact implies that the essential dynamical information relevant to CLRM may be represented with a model that is considerably less complex than the one originating from the discretization of the governing PDEs, and the definition of what we consider to be essential becomes one of the most relevant challenges for the research reported in this dissertation. When generating these reduced-complexity reservoir models, the first step is to define what information of the dynamical models should we reject while still achieving high performance of the CLRM, and what are the dynamic and static features of reservoir models that should be retained during the simplification process? The notions of reduced-order modeling for performance optimization connect to the data-based modeling for feedback control systems. Van den Hof and Schrama [1995] and Hjalmarsson et al. [1996] have given an overview on methods for closedloop system identification where the modeling criteria are control-relevant objective functions. In this dissertation, we call a reservoir model control-relevant, the one that is relatively faster than the full reservoir simulator, can be validated from data and embeds knowledge of the uncertainty space in its parametric description, and focus on the reservoir variables relevant for dynamical optimization. The generation of controlrelevant reservoir models raises many technical fundamental questions particularly on the definition of the fundamental variables, geological structures and dynamical behavior required for model-based operations. In this dissertation, the controlrelevant features of reservoir models are comprehended from two different, but not exclusive perspectives: structural and qualitative. The structural perspective for control-relevant models From the structural perspective (systems and control), the controllability and observability properties determine to a large extent the degree of influence over grid-block states from the controls, and the degree of information about grid-block states that we are able to infer from the measurements. In a few words, controllability clearly

33 12 Introduction characterizes the potential flow regimes that can be achieved in the oil field for a certain well configuration, and observability characterizes the spatial locations where the flow variables (saturation, pressure, etc.) can be effectively reconstructed from data. Due to the limited number of actuation and measuring devices at well locations, reservoir models are neither controllable nor observable, see, e.g., Van Doren et al. [2013]. However, system theory has been used to assess these properties for the 1D and 2D flow through porous media models, see, e.g., Zandvliet et al. [2008] and Van Doren et al. [2013]. Two-phase reservoir models are nonlinear, for instance the controllability and observability linear analysis has to be performed for short time horizons. It is possible to construct local approximations of controllability and observability Gramians, and to conclude about the most controllable and observable states. Van Doren [2010] has applied classical balancing techniques to derive reduced order models, where a new description for the state space is found such that the states of the dynamical system in the new coordinates are equally controllable and observable, i.e., the controllability and observability gramians are equal and the states are ordered according to their relevance and contribution to the I/O behavior, which is reflected in the Hankel singular values of the linearized system. Based on a case study, Van Doren et al. [2013] have concluded that two-phase flow dynamics lies in a subspace of the state space of lower dimension than the one originated from the discretization. In addition, it was observed that the pressures are highly controllable and observable around the wells and that the saturations are highly controllable around the oil-water front and observable around the wells. Jansen et al. [2009b] have studied the dynamics and controllability of the front of two immiscible fluids with identical viscosities in homogeneous porous media. The assumption of immiscibility implies that the steady state equation for pressure is linear with a diffusive nature, however, the front dynamics is described by a nonlinear model that depends explicitly on the velocity field and it is driven by pure convection. As a summary, the studies on structural analysis of reservoir models have focused on: 1. Determining the achievable flow regimes and fronts by performing a controllability analysis. 2. Determining the spatial location of the states and petrophysical properties which can be best inferred from the measurements, by performing observability analysis. The qualitative perspective for control-relevant models In reservoir engineering, the application of reservoir simulation can be extensively found in field development and planning, reservoir management, well location design and performance evaluation of reservoirs. For instance, from a qualitative perspective (reservoir engineering), the control-relevant models should be able to provide accurate prediction of variables relevant to the operation of reservoir assets, and

34 1.6 Control-relevant reservoir models 13 therefore what we consider to be control-relevant must be connected to the dynamical and economical performance of the oil reservoir. From the production optimization point of view, the use of Net Present Value (NPV), i.e., a measure of the cash flow for a given production strategy, generated at the end of a certain production period, could serve as a natural basis for a control relevant feature of the reservoir models. If we would restrict attention to life cycle production optimization under predefined production strategy and well configurations, this might be true. However when reservoir models are to be used also for testing new production strategies, as well as well-placement, infill-drilling and recompletion plans, the NPV measure is considered to be too coarse to distinguish between essential dynamic properties of the models. It is well known that the NPV is not able to infer the relevant aspects of the reservoir flow patterns associated with a particular production strategy, in other words: two essentially different geological models could lead to the same NPV under similar production strategies, but on the basis of essentially different flow patterns. Other variables to be considered as control-relevant features are the oil and water rates at the production wells. Although these measures are less coarse than the NPV, oil and water production rates (combined with pressure measurements) only provide local information of the flow behavior around the wellbore, which cannot be extrapolated to characterize the flow performance of spatial locations far from the production sites. Therefore it suffers from similar limitations as measures based on NPV. While streamline simulators have been used to generate a fast characterization of the cumulative production rates, see, e.g., Park and Caers [2007], Scheidt et al. [2009] and Scheidt and Caers [2009], they have also led to a technique called dynamic fingerprinting see, e.g., Yeh et al. [2014], where streamline information (time-of-flight (TOF) or drainage time) is used to generate flow patterns that are, like a fingerprint, unique to each realization. One may consider the fingerprints control-relevant features to be preserved, though they are merely simplified descriptors of the much more complex spatial-temporal reservoir flow patterns in terms of the temporal evolution of the flow variables (phase saturations, pressures, etc.) for a particular production strategy. Reservoir flow patterns are numerical solutions of the pressure and transport partial differential equations (PDEs) (Aziz and Settari [1979]), and they represent the temporal evolution of the dependent variables in the spatial domain of the reservoir. Therefore, the discrete-time trajectories for pressure, saturation, temperature, etc., are usually large-scale data structures, with dimensions induced by the number of grid cells of the reservoir model. Particularly, when working with an ensemble of realizations, the set of reservoir flow patterns for a particular strategy is used to generate robust strategic decisions on how to operate the oil field in a profitable way.

35 14 Introduction Discussion The structural and qualitative perspectives to comprehend the control relevant features of reservoir models are not contradicting and have elements in common. The qualitative perspective considers the reservoir flow patterns as the driving mechanism and the control-relevant feature for the development of strategies which leads to the optimal operation of the oil reservoir, while in the structural perspective, the control-relevant features are defined by the class of possible reservoir flow patterns than can be achieved with a certain well configuration. We have elaborated on the multivariable nature of the reservoir flow patterns: the dependent variables for reservoir simulation are usually the phase saturations, pressure, temperature, geomechanical stress, etc., and there is a variety of time scales and units involved in their description. In addition, the reservoir flow patterns have a spatial-temporal nature, which allows the use of multidimensional representations in a tensor structure. The idea of exploiting the structure of the flow variables and petrophysical properties in order to find low-complexity models for CLRM is described in the next sections. 1.7 Research objective and directions Previously, it has been deliberated that the control and optimization loops in the CLRM approach are constrained by the inherent complexity of the reservoir models, and that there are several technical and practical issues which limit the use of reservoir models for model-based operation of oil reservoirs. Hence, the research objective of this dissertation will be: The development of methods for reduced-complexity, control-relevant reservoir models, which can be validated from data. We limit the scope of this dissertation to the notions of the dynamical and static complexity discussed in the previous sections, where the state dimensionality and the current parametric description of the reservoir models restrict the use of large scale reservoir models in the CLRM, where the parametric uncertainty is described in the form of an ensemble of geological realizations. In order to generate reducedcomplexity models, we will exploit the spatial correlations of the variables that are control-relevant to the CLRM, and of the petrophysical parameters of the reservoir. Classical methods for low-complexity modeling do not consider the Cartesian structure of the physical variables in their procedures. To overcome the limitations of the classical methods, we introduce in this dissertation a spatial-temporal approach for pattern analysis. The multidimensionality of the reservoir flow patterns and petrophysical properties would allow a tensor representation, i.e., in multidimensional arrays. This creates a clear separation in time, space, variables and realizations which can be exploited for pattern analysis. Tensor decompositions and tensor analysis

36 1.8 Thesis outline 15 constitute a largely unexplored subject in reservoir engineering. For the characterization of geological parameters, Afra and Gildin [2013] andafra et al. [2014] have performed low-rank approximations of permeability fields using a tensor decomposition, and Gildin and Afra [2014] have used these representations for efficient history matching. For the reduction of dynamical complexity of reservoir models, we have utilized this framework for constructing reduced-order dynamic models, see, e.g., Insuasty et al. [2015b] and for the characterization of control-relevant flow profiles, see, e.g., Insuasty et al. [2015a, 2017b]. Hence, spatial-temporal tensor techniques appear to be an emerging technology to analyze patterns in reservoir flow patterns and to reduce the complexity of reservoir models. And therefore, the specific research question to be answered in this thesis can be formulated as: Can spatial-temporal tensor methods be exploited for developing reservoir models, that can be effectively used for model-based operations of oil reservoirs? In the reservoir community, the systematical handling of multidimensional objects, with spatial, temporal, and multivariable information has not been addressed. In this dissertation, we use a multidimensional approach to find tensor representations and decompositions of flow related variables and reservoir parameters. Tensor techniques will allow the construction of reservoir models with both reduced dynamic and static complexity while preserving the control-relevant aspects of the models, which are characterized by the spatial-temporal evolution of the reservoir flow patterns. The combination of iterative (large-scale) optimization, and history matching, with the need to use multiple model realizations makes CLRM into a computationally very demanding process for realistically-sized reservoir models, and the tensor approach for reservoir modeling aims to introduce a natural framework for simplifying the models used in the loops of the CLRM scheme. 1.8 Thesis outline This dissertation explores the use of tensor representations and decompositions in three areas of the mathematical modeling of oil reservoirs. Tensor methods for flow characterization In reservoir engineering, it is attractive to characterize the difference between reservoir models in metrics that relate to the economic performance of the reservoir as well as to the underlying geological structure. In Chapter 3, we develop a dissimilarity measure that is based on reservoir flow patterns under a particular operational strategy. To this end, a spatial-temporal tensor representation of the reservoir flow patterns is used, while retaining the spatial structure of the flow variables.

37 16 Introduction This allows reduced-order tensor representations of the dominating patterns and simple computation of a flow-induced dissimilarity measure between models. The developed tensor techniques are applied to cluster model realizations in an ensemble, based on similarity of flow characteristics. In Chapter 4, we apply the tensor methodology for flow characterization and model clustering to define reduced-size ensembles, which are used to accelerate model-based operations (robust production optimization and data assimilation). Tensor methods for the reduction of dynamical complexity In Chapter 5, a novel framework for reduced order modeling in reservoir engineering is introduced, where tensor decompositions and representations of flow profiles are used to characterize empirical features of flow simulations. The concept of classical Galerkin projection is extended to perform projections of flow equations onto empirical tensor subspaces, generating in this way, reduced order approximations of the original mass and momentum conservation equations. Tensor methods for the reduction of static complexity In Chapter 6, we use tensor methods to represent and approximate petrophysical properties. The efficiency of tensor approximations when representing channelized structures is verified as they require only 0.6% of the original amount of information to reconstruct geological structures appropriately, while a PCA approximation requires almost 63% (1 order of magnitude higher). In order to check the potential use of the tensor approach for parameter estimation, our experiments involve the estimation of the permeability field of a channelized reservoir using three alternative parametrizations: grid block, low-dimensional PCA and tensor-based. The uncertainty space is covered by an ensemble of channelized realizations which are updated every two years. After the final history match, the updated ensemble is used to predict total rates, pressures and water breakthrough. For the case of grid block property updates, an excellent history match is observed (over-parameterization), however, the ensemble has poor prediction capabilities due to a loss of geological realism. When using tensor representations, we observe that the geological realism of the updated ensemble is kept while having prediction capabilities similar to the one achieved with the PCA approach.

38 2 CHAPTER Reservoir modeling, optimization and algebraic techniques 2.1 Introduction Mathematical modelling is a process which uses mathematics as a language to describe the governing interactions of physical phenomena. As any cognitive process, it can be considered as an art which uses intuition, thoughts and experiences to create knowledge. However, mathematical modelling has as much of an art as of science, and it has the structure of scientific method. Typically, the development of mathematical models of physical processes can be divided into three stages, see, e.g., Hangos and Cameron [2001]: mathematical formulation, experimental validation and performance evaluation: 1. Mathematical formulation: In the first stage, the physical laws that describe the relations between the process variables and model parameters are determined. For continuous time and spatially distributed parameter systems, the use of (partial) differential equations has become a standard practice as it allows the representation of infinitesimal variations of physical variables in time and space. This stage includes the definition of boundary conditions and the types of exogenous signals that may act as controls and disturbances to the system. 2. Experimental validation: Typically, the model parameters are unknown. In the second stage, measurement data is used to calibrate model parameters in such a way that the outputs of the mathematical model are able to match the experimental measurements. Usually, estimation and optimization routines are used to find the set of model parameters that minimize the mismatch between the field measurements and model outputs. 3. Model-based process operation: In the third stage, control theory and algebraic techniques are used to measure and increase the dynamical performance of engineering systems using mathematical models. 17

39 18 Mathematical preliminaries In this chapter, we introduce the models and the estimation, optimization and algebraic techniques to be used in this thesis. 2.2 Reservoir models This section concerns the first stage of the mathematical modeling of oil reservoirs, i.e. the mathematical formulation. In this dissertation, we focus on the dynamical models to be used in the waterflooding stage, where we study the phenomena of two-phase (oil, water) flow through porous media, see e.g. Aziz and Settari [1979], Chavent and Jaffre [1986], Ertekin et al. [2001], Ewing [1983], Jansen [2013]. The target variables are phase saturations and pressure, the fluid properties are the densities and viscosities, and the petrophysical parameters to be considered are the permeability, and rock porosity. As pointed out in Chapter 1, the transport equations are characterized by the mass and momentum balances of the fluid phases. The boundary conditions consist on zero perpendicular flow through the boundaries, which for the sake of simplicity will be omitted in the description. The mass balance for the phases in an infinitesimal volume leads to the following differential equations: ρ w φs w + (ρ w v w ) ρ w q w = 0 t (2.1) ρ o φs o + (ρ o v o ) ρ o q o = 0, t (2.2) where is the divergence operator, s is the phase saturation which is a function of time and space, t is time, ρ is the phase density, φ is the rock porosity, q is a volumetric source, v is the phase velocity, and the subscripts w, o denote for water and oil phases respectively. The first term of the equations (2.1) and (2.2) correspond to the mass accumulation, the second terms corresponds to in inflow of mass due to the phase velocity fields and the third term corresponds to source terms. The velocity fields in (2.1) and (2.2) are determined by Darcy s law, which is a simplification of the momentum balance under the assumption of slow motion of the phases: v w = kw r µ w K( p w ρ w ge 3 ) (2.3) v o = ko r µ o K( p o ρ o ge 3 ), (2.4) where e 3 is the 3th standard unit vector in R 3, p is pressure which is a variable of time and space, g is gravity, z is the depth coordinate, µ is the phase viscosity, K is the rock permeability and k r is the relative permeability, which is a phenomena induced by the interaction of different phases in motion, and reduces the mobility of the different phases displacing through the porous media. The expressions for the

40 2.2 Reservoir models 19 relative permeability induce a strong nonlinear dependency of the velocity fields in (2.3) and (2.4) on the water saturation: where k w r = k w r0s nw (2.5) k o r = k o r0(1 S) no, (2.6) s w s con w S = 1 s w s r s con, (2.7) w k r0 is the end-point relative permeability, n o and n w denote the Corey exponents for oil and water saturations, s con w and the s r are the connate water and the residual oil saturations respectively. Typical curves for relative permeability are shown in Fig Figure 2.1: Relative permeabilities. The mass and momentum balances in Eqs. (2.1) to (2.4) constitute a system of 4 equations with 6 unknowns, therefore we require the following closure equations for the system to be solvable: s w + s o = 1 (2.8) p w p o = p c (s w ), (2.9) where p c (s w ) is the capillary pressure function. In this thesis, we assume that the capillary effects are negligible (p c (s w ) = 0) and incompressible flow, i.e. there is no dependency of the phase densities ρ w, ρ o on the pressure. Combining Eqs. (2.1) to (2.4) and (2.8) to (2.9) yields:

41 20 Mathematical preliminaries ρ w φ (1 s o) t [ k w ] r + ρ w K( p ρ w g z) ρ w q w = 0 (2.10) µ w ] ρ o φ s o t + [ ρ o k o r µ o K( p ρ o g z) ρ o q o = 0, (2.11) which is a system of nonlinear PDEs with the oil saturation s o and pressure p as variables. The nature of the model in Eqs. (2.10) and (2.11), indicate that the pressure has a diffusive behavior, while the oil saturation has a diffusive-convective behavior do to the hyperbolic nature of the equations, see e.g. Aziz and Settari [1979], Jansen [2013] State space representations In order to derive numerical solutions for the reservoir model in (2.10) and (2.11), we use spatial discretization techniques. Gridding of the spatial domain leads to a set of ordinary differential equations for the oil saturation and pressure for every grid cell, and to a state space formulation of the reservoir model. The Finite Volume (FV) technique, see, e.g., Aziz and Settari [1979], Eymard et al. [2000], Fung et al. [1992], Rozon et al. [1989], is the standard method for model discretization in the reservoir community, as the discrete version of the reservoir models have a clear physical interpretation while still preserving balance equations. In order to illustrate the procedure, we apply the FV method to the diffusive term of the 1D version of (2.11). Let us consider an uniform grid over the spatial domain, where the nodes are labeled from 1,, N, and let us apply the FV method at the ith grid node. The FV method starts with the integration of the model over a finite control volume around the grid i: x d ( dx kr o ρ o K dp ) ( dx = ρ o A ko r K dp ) ( ρ o A ko r K dp ), (2.12) µ o dx µ o dx i+ 1 µ 2 o dx i 1 2 where A is the cross-section area of the control volume face and x is the volume. The expression in Eq. (2.12) has a clear physical interpretation as it corresponds to the inflow and outflow of diffusive flux of p over the east (i ) and west (i 1 2 ) faces of the control volume of the ith grid node. Now, these fluxes are approximated using a finite difference scheme as follows: ( ρ o A ko r K dp ) µ o dx ) ( ρ o A ko r µ o K dp dx i+ 1 2 i 1 2 Γ i+ 1 2 Γ i 1 2 ( pi+1 p ) i x ( pi p i 1 x (2.13) ), (2.14)

42 2.2 Reservoir models 21 where the values of Γ are the average values of the petrophysical and fluid properties of the grid point i and its neighbors: Γ i+ 1 [ = 1 ( ) ρ 2 o A ko r K 2 µ o Γ i 1 2 = 1 2 i ] ( ) + ρ o A ko r K µ o i+1 ] [ ( ) ( ) ρ o A ko r K + ρ o A ko r K µ o i µ o i 1 (2.15). (2.16) By combining Eqs. (2.13) to (2.16), the volume integration on Eq. (2.12) can be approximated as: x ( d ( dx kr o ρ o K dp ) Γi+ 1 2 dx µ o dx x + Γ 1 i 2 x ) p i + ( ) Γi 1 2 p i 1 + x ( ) Γi+ 1 2 p i+1. x (2.17) A similar FV analysis can be performed for the accumulation term of the model equation (2.11), leading to: x ρ o φ s o t dx 2 xρ oφṡ i + xρ o φṡ i 1 + xρ o φṡ i+1, (2.18) where the upper dot denotes for derivative operator with respect to time. By applying the forward discretization operator for finite differences to the time derivative term, we obtain: ( ) δ f ρ o φ s x t dx 2 xρ ] oφ [(s i+1 2s i + s i 1 ) k+1 (s i+1 2s i + s i 1 ) k, (2.19) t where δ f is the forward difference operator and t is the time step. The FV representation of the source terms are omitted for the sake of simplicity. The expressions in Eqs. (2.17) and (2.19) combined with the source terms define a set of nonlinear ordinary differential [ equations for the ith grid-block node. Let us define the vector of p states as x =, where p, s are the vectors of grid-block pressures and saturations, s] then the state space representation of the reservoir model has the form of: E(x k )x k+1 = A(x k )x k + B(x k )u k, (2.20) where x k and u k are the states and the control inputs at time k < K, E(x k ) contains the parameters of the accumulation term, A(x k ) relates to the parameters of the diffusive term and u k contains the source elements. The expressions in Eqs. (2.17) and

43 22 Mathematical preliminaries (2.19) indicate that E(x k ), A(x k ), B(x k ) are sparse matrices, and for the 1D example consider in this section, the matrices E(x k ), A(x k ) have a tridiagonal structure, see e.g., Jansen [2013]. The state space model in Eq. (2.20) fits into the category of linear time-variant systems (LTV), as there is a nonlinear dependency of the system matrices on the states Reservoir simulator The model in (2.20) can be written in its residual formulation: g(k, x k, x k+1, u k ) =E(x k )x k+1 A(x k )x k B(x k )u k = 0, (2.21) d sim k = h(k, x k, u k ) (2.22) where g is a real vector-valued nonlinear function of the states and controls, and h is the real-valued vector nonlinear function for the outputs d sim. Classical numerical methods for solving systems of nonlinear equations can be used to find the state vector x k+1 that satisfy Eq. (2.21). For this, the Newton-Raphson method, see e.g. Ben-Israel [1966], Coats et al. [1998], Young et al. [1983], Ypma [1995], finds iteratively the zeros of the residual operator g(k, x k, x k+1, u k ), which for the sake of simplicity we call g k. The method requires the solution of the following linear systems of equations: g(k, x k, x n+1 k+1, u k) = g ( ) k x n+1 x n+1 k+1 xn k+1, (2.23) k+1 where the superscript n counts for the Newton iterations. Intuitively, the method approximates g k by its tangent function characterized by the Jacobian g k, and x n+1 k+1 given the initial condition x k, it approximates the zeros of g k, i.e. the state vector x k+1 by the intersection of the tangent function with zero. According to Chen et al. [2006], for large-scale reservoir models ( 10 5 states), about 80% 90% of the total simulation time is spent on solving the linear system in Eq. (2.23). 2.3 Optimization methods for model-based operations of oil reservoirs Introduction This section concerns the experimental validation and model-based operation stages of mathematical modeling.in this dissertation, the reservoir models described in the previous section are used for the optimization and parameter estimation loops of the

44 2.3 Optimization methods for model-based operations of oil reservoirs 23 CLRM, which can be defined mathematically as optimization problems. Generally, and without considering state, input and parameter constraints, the optimization problems in production optimization and parameter estimation can be formulated as follows: min u,θ J(x 1,..., x K, u, θ) (2.24) subject to: g(k, x k, x k+1, u k, θ) = 0, (2.25) where u = [u 1,, u K ] the collection of control inputs, θ = [θ 1,, θ N ] the collection of all the grid-block parameters, g is the residual form of the reservoir simulator defined in Eq. (2.21), and J is an objective function that relates to a financial measure (e.g. NPV) for the case of production optimization, and relates to the misfit between measurements and model outputs for the case of parameter estimation The adjoint equations and gradients With the application of variational methods and optimal control, see e.g. Bellman [1956], Pontryagin [1987], Struwe and Struwe [1990], we augment the objective function J in (2.24) with the dynamic constraint (2.25) in order to find the necessary conditions for optimality. For applications of optimal control theory in reservoir engineering see e.g. Sarma et al. [2005, 2006] and Jansen et al. [2008]. With the introduction of Lagrange multipliers, the augmentation of the cost function leads to: Ĵ = J + N λ k g(k, x k, x k+1, u k, θ), (2.26) k=1 where λ k is the kth Lagrange multiplier. Now, the effect of infinitesimal state dx k, control du k and parameter dθ perturbations on the augmented cost function can be found by applying the chain rule of differentiation: dĵ =dj + N + N k=1 k=1 λ k λ k g k x k dx k + g k u k du k + N k=1 N k=1 λ k λ k g k x k 1 dx k 1 (2.27) g k θ dθ changing the index of the terms that involves x k 1 we get

45 24 Mathematical preliminaries and regrouping terms dĵ =dj + N + N k=1 k=1 λ k λ k N 1 g k dx k + λ g k+1 k+1 dx k (2.28) x k x k g k u k du k + N k=1 k=0 λ k g k θ dθ ( N dĵ =dj + k=1 λ k g k + λ g k+1 k+1 x k x k + λ 1 ) dx k + N k=1 λ k g k u k du k + N k=1 λ k g k θ dθ g 1 dx 0 λ g N+1 N+1 dx N (2.29) x 0 x N Similarly, the infinitesimal variation of the cost function dj under infinitesimal state dx k, control du k and parameter dθ perturbations is found by applying the chain rule of differentiation: dj = N k=1 J x k dx k + and combining it on (2.29) we get that N k=1 J N θ dθ + J du k (2.30) u k k=1 ( N dĵ = + k=1 ( N k=1 + λ 1 λ k λ k g k + λ g k+1 k+1 x k x k ) g k + J u k u k ) + J dx k x k ( N du k + k=1 g 1 dx 0 λ g N+1 N+1 dx N, x 0 x N λ k ) g k θ + J θ dθ (2.31) The necessary conditions of optimality requires stationarity of dĵ for all the perturbations, which leads to the following set of equations: λ 1 λ N+1 g 1 = 0 x 0 (2.32) g N+1 = 0 x N (2.33)

46 2.3 Optimization methods for model-based operations of oil reservoirs 25 λ g k+1 k+1 x k λ g k k + J u k λ g k k θ + J θ + λ k g k + J = 0, (2.34) x k x k u k = 0, (2.35) = 0. (2.36) The temporal evolution of the Lagrange multipliers is described by the dynamical system in (2.34), which is known as the adjoint model. The Lagrange multipliers λ N+1 are calculated using (2.33), and the adjoint model is run backwards in time. The fully implicit reservoir simulators as presented in Eq. (2.21) usually provide the Jacobian of the residual operator with respect to the states g k x k and g k+1 x k, see e.g. Sarma et al. [2005]. The expressions in Eqs. (2.35) and (2.36) will be used to compute gradients of objective functions in the next sections Production Optimization The objective of production optimization is the design of a production strategy that maximizes the profits at the end of the life cycle of the reservoir. Traditionally, the NPV has been used as a measure to determine the financial performance, and can be understood as the net cash flow during the operational life cycle of the reservoir: J = K k=1 [ Ninj i=1 r wi (u wi,i ) k + ] N prod ] j=1 [r wp (y wp,i ) k + r o (y o,j ) k t (1 + b) t k k τ (2.37) where u wi,i are the injection rates (control inputs), K is the optimization horizon, t k is the time step, b is the discount factor, N inj and N prod are the number of injectors and producers, r wi and r wp denote the cost of water injection and production, r o represents the price of oil produced and y wp,i and y o,j denote the water and oil production rates (measured outputs). In this dissertation, optimal control theory and the corresponding adjoint methods are used to compute optimal strategies that maximizes (2.37), see Jansen [2011] for a complete review. For production optimization, the model is considered to be known, and there is no explicit dependency of the objective function J on the model parameters θ. For instance, the condition for optimality in (2.36) is not considered, and only (2.35) is used to compute gradients of J with respects to the controls u k. (2.35) can be interpreted as the infinitesimal variation of the augmented cost function Ĵ due to infinitesimal variations of the controls, and non-optimal control strategies will lead to a value different than 0 in (2.35). The gradients of Ĵ with respect to the controls are given by: Ĵ u k = λ k g k u k + J u k. (2.38)

47 26 Mathematical preliminaries From the implementation point of view, the term J u k is easily computed, however the term g k u k requires a considerable amount of work, as usually millions of lines of code are required to implement this Jacobian in a computer program. Currently, automatic differentiation (AD) solvers have been successfully implemented in some reservoir simulators like the MRST (Matlab Reservoir Simulation Toolbox - SINTEF), see e.g. Lie et al. [2012], and the AD-GPRS (Automatic Differentiation General Purpose Research Simulator - Stanford). With (2.38) at hand, we can use any gradientbased numerical method to find the optimal solution u k. Robust Optimization Decision making processes in the presence of model uncertainty constitute a major challenge in reservoir engineering and closed-loop reservoir management. In order to design a production strategy to account for parametric uncertainties in reservoir models, Robust Optimization (RO) has been applied, see Capolei et al. [2015], Chen et al. [2012], Fonseca et al. [2015], Siraj et al. [2015], Van Essen et al. [2009], Yasari et al. [2013]. In Robust Optimization, we maximize the average value of NPV over an ensemble of R realizations. with gradient J rob = 1 R R J i, (2.39) i=1 dj rob du k = 1 R R i=1 dj i du k, k = 1, 2,, K, (2.40) where u k and y k are the input and output vectors at time step k. RO requires the computation of the numerical gradients of the cost function for every member of the ensemble and the technique can become computationally prohibitive for big ensembles of large-scale models due to the burden associated with the computational load required to compute numerical gradients and simulations. Approximation of the gradients can be done using ensemble optimization, see e.g. Chen et al. [2009a], Fonseca et al. [2014b,c, 2015], without the need of an adjoint formulation, however, this approach is even more computational expensive than gradient-based methods for large ensembles of reservoir models Parameter estimation The objective of the parameter estimation loop in CLRM is to assimilate with the model the information contained in the measurements taken during the operation of the oil reservoir, which typically are the production rates and bottom-hole pressures at the well locations. This has been a very prolific field of research and many

48 2.3 Optimization methods for model-based operations of oil reservoirs 27 algorithms have been developed in the last decades. On the one hand, the variational methods or gradient-based approaches use the adjoint equations to compute gradients of objective functions, see e.g. Chavent et al. [1975], Chen et al. [1974], Lee and Seinfeld [1987], Oliver et al. [2008], Wasserman et al. [1975], Watson et al. [1980], Zhang and Reynolds [2002]. These methods aim to minimize the mismatch between the model output and the data, see e.g. Talagrand and Courtier [1987], with an additional regularization term to account for the prior information: J(θ) = 1 2 (dsim (θ) d) C 1 d (dsim (θ) d) (θ θ prior) C 1 prior (θ θ prior), (2.41) where d is the vector of measurements, θ prior is the initial set of rock properties which keeps the geological realism and C d and C prior are the covariance matrices of the data noise and prior parameters. For parameter estimation, there is no dependency of the objective function J on the controls u k, for instance the condition for optimality in (2.35) is not considered, and only (2.36) is used to compute gradients of J with respects to the model parameters θ. (2.36) can be interpreted as the infinitesimal variation of the augmented cost function Ĵ due to infinitesimal variations on the model parameters. The gradients of Ĵ with respect to θ are given by: Ĵ θ = g k λ k θ + J θ. (2.42) The implementation of the gradient g k θ in (2.42) poses similar challenges as the implementation of the gradient for production optimization, and alternative (ensemble) methods which do not require the computation of gradients have been developed. The Kalman Filter (KF), see, e.g., Grewal [2011],is based on Bayesian estimation principles and uses the reservoir model to forecast the states and estimate the probability distribution of the model parameters with the aid of measurement data. The KF assumes the states, parameters and measurements to be uncertain with associated Gaussian probability distributions, and uses the available data to update the posterior states and parameters distributions, such that a likelihood function similar to (2.41) is minimized. In order to define the basic notions of the KF, let us use the model description in (2.21) and let us assume that u k = 0 for the sake of simplicity. The KF is suited for linear systems, and in order to illustrate the algorithm we use the linearization of the model in (2.21). The KF for state estimation can be formulated as a sequence of prediction and estimation steps as follows: 1. Initialization: Under the assumption that the state uncertainty has a Gaussian distribution, the vector of true states x t : x k t N (x f k, C k f ), (2.43) where x f k are the predicted states vector from the simulation model, and C k f is the states covariance matrix.

49 28 Mathematical preliminaries 2. Forecast: Considering the state uncertainty, the true states are governed by the linearized model: x t k+1 = [ gk+1 ] 1 gk+1 x f k+1 x f k x f k + n k, (2.44) d k = h k x f x f k + r k, (2.45) k where n k N(0, Q k ) and r k N(0, R k ) model the state and measurements uncertainty. Then, the state forecast and its covariance matrix can be defined as: x f k+1 = E[x t k+1] = [ gk+1 ] 1 gk+1 x f k+1 x f k x f k, (2.46) C f k+1 = E[(x t k+1 x f k+1)(x t k+1 x f k+1) ] ( [ gk+1 ] 1 gk+1 [ f gk+1 = C k x f k+1 x f k x f k+1 ] 1 gk+1 x f k ) + Q k. (2.47) 3. Assimilation: In this step, the Kalman gain is defined as: ( [ f h ) k+1 K k+1 = C k+1 x f k+1 h k+1 x f k+1 ( ] f h ) 1 k+1 C k+1 + Rk x f. (2.48) k+1 The Kalman gain is now used to propagate the covariance matrix for the predicted states according to: C a k+1 = E[(x t k+1 x a k+1)(x t k+1 x a k+1) ] ( h ) ( k = I K k+1 x f C f h ) k k+1 I K k+1 + Kk+1 k x f R k K k+1, (2.49) k and the measurements are assimilated to propagate the probability distribution of the states using the Kalman gain: ( x a k+1 = x f k+1 + K k+1 d k h k+1 x f x f k+1 k+1 ), (2.50) The KF description used as illustration in this section uses the Jacobians of the reservoir model, and it is known as the Extended KF, see, e.g., Jazwinski [2007]. As it has been discussed before, obtaining the Jacobian matrices of dynamical systems

50 2.3 Optimization methods for model-based operations of oil reservoirs 29 can be a very difficult task, and ensemble-based filtering techniques for computerbased history matching have been successfully applied in the reservoir, meteorology and oceanography communities, see, e.g., Aanonsen et al. [2009], Bertino et al. [2003], Bianco et al. [2007], Burgers et al. [1998], Chen et al. [2009b], Dong et al. [2006], Evensen [2003, 2009]. For overviews we refer to the textbook by Oliver et al. [2008], and the review paper of Oliver and Chen [2011]. Here we will use the Ensemble Kalman filtering method for which we refer to the textbook of Evensen [2009] and the review paper of Aanonsen et al. [2009]. The Ensemble KF (EnKF) approximates the nonlinear filtering problem by taking samples of the PDFs of the states and use them to propagate the statistical properties of the posterior estimates, see, e.g., Evensen [2003], while avoiding computing the Jacobians of the dynamical system with respect to the parameters and states. In order to provide a clear illustration on how to perform state and parameter estimation simultaneously using EnKF, let us use a more explicit formulation of the reservoir model: x k+1 = f(x k, u k, θ), (2.51) where f is a vector-valued nonlinear function of the states x k, the petrophysical parameters θ and the inputs u k. Let d k+1 sim = h(x k, u k ) be the nonlinear output operator, and let us consider the augmented state vector z k : with augmented output vector y k : x k+1 f(x k, u k, θ) z k+1 = θ = f(z k, u k ) = θ, (2.52) d sim k+1 h(x k, u k ) y k+1 = Hz k+1 = [ 0 0 I ] f(x k, u k, θ) θ. (2.53) h(x k, u k ) The EnKF approach follows the sequential steps of the classical KF, but uses an ensemble of R parameterizations of the realizations Ω = {θ 1, θ 2,, θ R } to update the mean and covariance of the augmented state z: 1. Forecast: For every model realization in the set Ω, the forecast of the ensemble of augmented states is generated, and the matrix Ξk+1 f = [ z 1,f k+1, z2,f k+1,, ] zr,f k+1 is constructed according to: z i,f k+1 = x i,f k+1 θ i,f k+1 d sim,i,f k+1 = k, u k, θ i,a k ) θ i,a k. (2.54) h(x i,a k, u k) f(x i,a

51 30 Mathematical preliminaries 2. Assimilation: The covariance matrix associated to the ensemble of augmented states is computed as: ( Ξ f C f k+1 = k+1 z f k+1 1 )( Ξk+1 f zf k+1 1 ), (2.55) R 1 R i=1 zi,f where z f k+1 = 1 R k+1 is the average of the augmented states of the ensemble and 1 is the column vector with all the elements equal to 1. The Kalman gain then is defined as K k+1 = C f k+1 H (HC f k+1 H + R k ) 1, with R k the covariance matrix of the measurement noise. The states z k are assimilated with the data using the Kalman equations as follows: z i,a k+1 = zi,f k+1 + K k+1(d k Hz i,f k+1 ), (2.56) In practice, the EnKF is implemented in a computationally more efficient fashion. Moreover it is often necessary to implement localization techniques to avoid spurious correlations. For details we refer to Evensen [2009] and Aanonsen et al. [2009]. 2.4 Tensor modeling and decompositions This section concerns the use of algebraic techniques for model-based operation of physical processes. The purpose of introducing algebraic methods to the modeling of oil reservoirs is to create reduced-order representations of the flow variables and models. The low-rank approximation of matrices is well understood, see, e.g., Eckart and Young [1936], Golub and Van Loan [2012], and there have been huge developments on the extension of low-rank approximations to multilinear arrays, see, e.g., Chen and Saad [2009], De Lathauwer et al. [2000a,b], De Silva and Lim [2008], Kolda and Bader [2009], Lim [2004]. In this section, we introduce the concepts of multilinear algebra used in this dissertation Fundamentals of linear and multilinear algebra Tensors are the multilinear extensions of vectors and matrices and can be represented as multi-way arrays, and similar to the matrix case, tensors can be interpreted as multilinear mappings. With this interpretation, a tensor is defined as a multilinear mapping S : R I1 R I2 R I N R. Let T N be the set of all multilinear mappings R I1 R I2 R I N R, then T N becomes a vector space when it is equipped with additive and multiplicative properties, see e.g. van Belzen [2011], Van Belzen and Weiland [2012]. Given S, R T N we have that: Additivity: Q = S + R T N, where the (i 1 i 2 i N )th element of Q, i.e. q i1i 2 i N = s i1i 2 i N + r i1i 2 i N, the sum of the (i 1 i 2 i N )th elements of S and R.

52 2.4 Tensor modeling and decompositions 31 Tensor multiplication by a scalar: P = αs T N for any α R, where the (i 1 i 2 i N )th element of P, i.e. p i1i 2 i N = αs i1i 2 i N. It is important to understand that the elements of S, i.e. s i1i 2 i N respect to a set of basis functions: are defined with {f i1 1, for i 1 = 1,, I 1 }, {f i N N, for i N = 1,, I N }. (2.57) If the vector space T N is equipped with an inner product, it allows the definition of a normed space. Let S, R T N, the inner product, between S, R is defined as: S, R TN := I 1 i 1=1 I N I 1 i N =1 j 1=1 I N j N =1 s i1i 2 i N r j1j 2 j N f i1 1, f j1 1 1 f i N N, f j N N N, (2.58) where, i is the inner product on R i. Hence, an induced tensor norm can be defined as S F = S, S TN, known as the Frobenious norm, which is equal to S F = I1 i 1=1 I N I 1 i N =1 j 1=1 I N j N =1 s i1i 2 i N 2, (2.59) for the special case when the basis functions in Eq. (2.57) are orthogonal Tensor decompositions and approximation The most general formulation for a tensor decomposition is provided by the Tucker decomposition, see e.g. the work of Tucker [1966] and Kolda and Bader [2009], where the tensor S is represented by the multiplication of the elements of a full core tensor with rank 1 tensors, defined by the basis functions for each of the tensor coordinate space. The Tucker decomposition can be defined as: S = I J i=1 j=1 k=1 K σ ijk ϕ i ψ j χ k = I J i=1 j=1 k=1 K σ ijk Θ ijk, (2.60) where the scalars σ ijk R are the elements of the so called core tensor of size I J K and where Θ ijk := ϕ i ψ j χ k is the outer product of vectors ϕ i R I, ψ j R J and χ k R K. This makes Θ ijk a rank-one 3-way tensor. A graphical illustration of the Tucker decomposition (2.60) is depicted in Figure 2.2. In such representation, the sets {ϕ i } I i=1, {ψ j} J j=1 and {χ k} K k=1 are usually taken as a basis of R I, R J and R K, respectively, and the Tucker decomposition (2.60) is viewed

53 32 Mathematical preliminaries Figure 2.2: Schematic description for the truncation of a Tucker decomposition of a 3D tensor. as a representation of the tensor with respect to these bases. More precisely, the rankone 3-way tensor Θ ijk is a multilinear mapping Θ ijk : R I R J R K R defined as Θ ijk (ϕ, ψ, χ) := ϕ i, ϕ ψ j, ψ χ k, χ which is a product of inner products in R I, R J and R K. At this stage, it is important to observe that the mapping Θ ijk, defined in this way, is linear in each of its argument. Moreover, if the bases {ϕ i } I i=1, {ψ j} J j=1 and {χ k} K k=1 are all orthonormal sets (that is, ϕ i, ϕ i = 1 if i = i and is zero otherwise for vectors {ϕ i } I i=1 ), it follows that S(ϕ i0, ψ j0, χ k0 ) = I J i=1 j=1 k=1 K σ ijk Θ ijk (ϕ i0, ψ j0, χ k0 ) = σ i0j 0k 0 (2.61) for any triple of indices (i 0, j 0, k 0 ) with 1 i 0 I, 1 j 0 J and 1 k 0 K. In words, this says that the entries of the core tensor represent the tensor S when evaluated at its (orthonormal) basis vectors. If the bases are orthonormal, then this observation naturally identifies the entries σ i0j 0k 0 of the core tensor with the evaluation of the tensor S at its (i 0, j 0, k 0 )th basis element. If the bases are non-orthonormal bases, then the multilinear functional S : R I R J R K R defined in (2.61) and operating over the vectors (ϕ, ψ, χ) changes its representation, and produces: S(ϕ, ψ, χ) = I J i=1 j=1 k=1 K σ ijk ϕ i, ϕ ψ j, ψ χ k, χ (2.62) A particular case of the Tucker decomposition is the canonical polyadic (CP) decomposition, also known as the PARAFAC decomposition, see e.g. Bro [1997], Harshman [1970], Harshman and Lundy [1994], has been successfully applied in the field of chemometrics. In the CP decomposition the core tensor is "diagonal", i.e., the elements of the core tensor σ ijk 0 for i = j = k, and σ ijk = 0 otherwise. For the Tucker decomposition, a low-rank approximation of S can be obtained by decomposing (2.60) according to S = Ŝ + S where, for Î I, Ĵ J, K K, the

54 2.4 Tensor modeling and decompositions 33 tensor Î Ĵ Ŝ := K i=1 j=1 k=1 σ ijk Θ ijk (2.63) is viewed as the approximation of S to its modal-rank (Î, Ĵ, K) truncation and where S := S Ŝ is viewed as the corresponding approximation error. The size of the approximation error is measured in Frobenius norm and satisfies S Ŝ F S F Ŝ F, (2.64) provided that the bases {ϕ i } I i=1, {ψ j} J j=1 and {χ k} K k=1 are orthonormal sets. The Frobenius norm of tensors in (2.64) is used to compute a relative approximation error related to the tensor approximation as follows: e rel = S Ŝ F S F. (2.65) Suppose that the above tensor decomposition is applied to the data corresponding to a two-dimensional rectangular saturation field that evolves over time. The oil saturation snapshot at time m, X m, is then represented as a matrix of dimension I J. A number of K samples X m is stored in an order-3 tensor S of size I J K. This tensor is approximated as in (2.63), and results in the approximate sample X m of the saturation field defined by the order-2 tensor Î Ĵ X m = Ŝ(,, e m) = K σ ijk χ k, e m ϕ i ψ j = i=1 j=1 k=1 i=1 j=1 Î Ĵ αij m ϕ i ψ j (2.66) where e m is the mth standard unit vector in R K and where αij m K := k=1 σ ijk χ k, e m are real-valued coefficients in the expansion (2.66) of (rank 1) two-dimensional fingerprints of the saturation field. The coefficient αij m is a linear combination of the mth element of the basis functions in the set {χ k } K k=1, i.e., αij m = K k=1 σ ijk χ (m) k, where χ (m) k = χ k, e m Algorithms for tensor decompositions Clearly, the approximation accuracy of (2.63) and (2.66) depends on the choice of basis vectors ϕ i, ψ j, and χ k, their ordering and the elements in the core tensor. There exist many algorithms to select these bases in such a way that the approximation error (2.64) is small or minimized.

55 34 Mathematical preliminaries Figure 2.3: Time snapshots for oil saturation and (10, 10, 10) approximation computed using TSVD, HOSVD, MSVM and SDM. The problem of finding these sets can be formulated as the optimization problem min {ϕ i} i Î,{ψj} j Ĵ,{χ k} k K S Î Ĵ K i=1 j=1 k=1 σ ijk ϕ i ψ j χ k F (2.67) which is to be solved subject to the constraint that the basis elements {ϕ i 1 i Î}, {ψ j 1 j Ĵ} and {χ k 1 k K} are orthonormal. This problem has an analytic solution only for the case where (Î, Ĵ, ˆK) = (1, 1, 1). For all other cases one has to resort to numerical approximations. Several algorithms have been proposed to compute tensor decompositions using orthonormal basis functions. The High Order SVD (HOSVD) proposed by De Lathauwer et al. [2000a] was the first extension of the classical SVD to the spatial-temporal case and the methodology is based on an unfolding procedure of tensors. The High Order Orthogonal Iteration (HOOI) by De Lathauwer et al. [2000b], the Tensor SVD proposed by Weiland and Van Belzen [2010], Maximum Singular Value Modal Rank (MSVM) and the Single Directional Modal-rank decomposition (SDM) by Shekhawat and Weiland [2014] compute singular values (elements of the core tensor) and basis vectors

56 2.4 Tensor modeling and decompositions 35 Tensor rank (Î, Ĵ, ˆK) TSVD HOSVD MSVM SDM (1, 1, 1) (3, 3, 3) (5, 5, 5) (7, 7, 7) (9, 9, 9) (11, 11, 11) (13, 13, 13) (15, 15, 15) Table 2.1: Relative approximation error of various algorithms for tensor decomposition. in a sequential way, where the singular values and vectors depend on a search direction at every decomposition level (Î, Ĵ, K). The tensor SVD, MSVM and SDM algorithms keep the tensor structure intact in such a decomposition procedure. In this dissertation, we consider the Tucker modal-rank type of decomposition, see Kolda and Bader [2009], which achieves orthonormal sets {ϕ i } I i=1, {ψ j} J j=1 and {χ k} K k=1. There are several tensor toolboxes available for the Matlab platform, like the Matlab Tensor toolbox, see Bader et al. [2015] and the Tensorlab, see Vervliet et al. [Mar. 2016]. In this dissertation, we have implemented the HOSVD and the SDM using tensor functions from the Matlab Tensor Toolbox, see, e.g., Bader et al. [2015]. In order to exemplify the performance of different algorithms, we have computed the tensor decomposition of a tensor S R I J K with I = 21, J = 21 and K = 100. We perform the truncation in (2.63) with (Î, Ĵ, ˆK) where Î = Ĵ = ˆK = r and 1 r 10. For different values of r, the relative approximation errors in (2.65) generated with different algorithms for tensor decompositions are presented in Table 2.1. We have found that some techniques are able to extract relevant information better than others. Both MSVM and SDM outperform the TSVD and their performance is comparable to the HOSVD. The slow rate of decrease of error in TSVD algorithm can be attributed to the restrictive space in which the algorithm searches it singular values, as it is a PARAFAC-type decomposition, see, e.g., Kolda and Bader [2009]. SDM, TSVD, MSVM perform better than HOSVD for the r = 1 approximation Computational complexity of tensor decompositions In this section, we address the computational complexity of tensor decompositions, with a particular emphasis on the HOSVD algorithm provided in De Lathauwer et al. [2000a]. To exemplify, let as consider a 3-way array S R I J K. The HOSVD algorithm flattens the tensor S over all of its modes, for instance, the flattened representation of S over the first mode is the matrix S 1 R I JK. Next we construct the sample covariance matrix associated to S 1, i.e. C 1 = S 1 S 1, where C 1 R I I. In

57 36 Mathematical preliminaries order to find the set of orthonormal basis functions for the first mode, we compute the SVD of C 1. The computational complexity of computing the SVD of an arbitrary matrix A R m n is of the order O(m 2 n + 4n 3 ), see, e.g., Golub and Van Loan [2012]. Hence, the computational complexity associated to the SVD of C 1 is O(5I 3 ). The HOSVD algorithm computes an SVD of all of the modal covariance matrices associated with the flattened tensor, for instance the HOSVD computation of the 3-way tensor S has a computational complexity of order O(5I 3 + 5J 3 + 5K 3 ). For the sake of illustration, we have performed an experiment to measure the time required to compute the HOSVD and the SVD of a tensor and matrix, with the same number of elements respectively. In this experiment, we consider a tensor of real elements S of dimensions and its flattened matrix representation S of dimensions , hence, S and S have the same number of elements. On the one hand, we use the HOSVD to compute a decomposition of S similar to the one in (2.60), and on the other hand, we calculate the SVD of S. The theoretical complexity of the SVD is of order O(10 12 ) while the complexity of the HOSVD is of order O(10 9 ). The experiment was performed on a personal computer equipped with an 8 Core Intel i7-3:4ghz and 32GB of RAM. The HOSVD of S and SVD of S were repeated 1000 times and the distributions of the computational time for each algorithm are depicted in Fig The experiment shows that the computation of tensor decompositions is faster than computing the SVD for algebraic structures with the same number of elements. This phenomena can be explained from the fact that it is usually faster to compute three SVD of covariance matrices of order I, J, and K independently, rather than computing one SVD of a matrix with I J rows and rank K. Another side of tensor computations is the reconstruction of a tensor S from its constitutive components in the decomposition in (2.60). More precisely, given the set of basis functions {ϕ i } I i=1, {ψ j} J j=1, {χ k} K k=1 and the core tensor with elements σ ijk, perform the algebraic operations described in (2.60) in order to compute S. The matrix equivalent to this operation S = UΣV, is the computation of the matrix S by performing the matrix multiplications given by the SVD. These algebraic operations are trivial from the mathematical point of view, however they play a role in the computational complexity of methods for large-scale tensors. For the experiment, given the tensor and SVD decompositions of the same S and S, we perform 1000 reconstructions for both tensor and matrix structures and the results are depicted in Fig The results shows that reconstructing the matrix S is one order of magnitude faster than reconstructing the tensor S. If we compare the results on Figs. 2.4 and 2.5, we conclude that the reconstruction of tensors from their spectral expansion consumes almost the same amount of time as the tensor decomposition itself.

58 2.4 Tensor modeling and decompositions 37 Figure 2.4: Distribution of the computational time for tensor and matrix decomposition algorithms. HOSVD of S and SVD of S were repeated 1000 times. Green: HOSVD. Blue: SVD Figure 2.5: Computational time for the matrix and tensor reconstruction methods. Reconstruction of S and of S from decompositions was repeated 1000 times. Green: HOSVD. Blue: SVD

59 38 Mathematical preliminaries 2.5 Concluding remarks In this chapter, we have elaborated on some of the theoretical concepts of the ingredients of CLRM: models, optimization, parameter estimation and algebraic techniques. The first three subjects have been summarized by several authors, see, e.g., Fonseca [2015], Kaleta [2011], Krymskaya [2013], Van Doren [2010]. We contributed on the incorporation of tensor methodologies into CLRM loops of production optimization and parameter estimation, and we have provided the definitions and algorithms for tensor decompositions required in this dissertation.

60 Part I A spatial-temporal approach for flow characterization 39

61

62 3 CHAPTER Tensor formulation of flow-based dissimilarity measures Introduction In the previous chapters, it has been described the inherent computational complexity of performing large-scale optimization and parameter estimation for reservoir systems. This complexity escalates when geological uncertainty is incorporated, and hundreds to thousands of model realizations of realistically-sized reservoir models are needed for the model-based operations of the CLRM. For this reason the selection of very few but representative reservoir models has become a relevant issue for the practice of reservoir engineering. Particularly, for the CLRM framework one would like to discriminate between realizations which are representatives of the different types of flow responses. For this reason, oil companies have used very few realizations which are often selected manually to achieve robustness in their operational strategies, see, e.g., Sarma et al. [2013]. When selecting representative models for CLRM from a large ensemble of realizations, models have to be screened based on their performance-related properties. In other words, there is a need for a dissimilarity measure between reservoir realizations that is relevant for model-based operation of oil reservoirs. There are several options for discriminating between model realizations, on the basis of either static or dynamic properties of the reservoir models. In Caers et al. [2010], Suzuki et al. [2008] the permeability fields have been used as a measure of dissimilarity. This has been done by defining a metric space to compare and cluster geological models that share common geological features. These dissimilarity measures based on static permeability or porosity properties are known to be quite different from measures applied to the dynamic behavior of the reservoir models, reflected in the corresponding flow patterns, as e.g. the evolution of oil saturation over the life cycle of the reservoir. 1 Substantial content of this chapter is also published or presented in E. Insuasty, P. M. Van den Hof, S.Weiland, and J.-D. Jansen. Flow-based dissimilarity measures for reservoir models: a spatial-temporal tensor approach. Computational Geosciences, pages 1 19,

63 42 Flow-based dissimilarity measures To exemplify this, upscaling of high resolution geological models, as is presented in Durlofsky [2005], shows that dissimilar reservoir models may have similar dynamical performance in terms of flow dynamics. At the same time, reservoir models that are close in geological properties can have essentially different flow patterns, and therefore different behavior from a dynamic operation point of view. In Chapter 1, we elaborated on the risky implications of selecting NPV as a controlrelevant feature and dissimilarity measure, as different geological realizations produce the same cash flow, under identical production strategies but on the basis of totally different flow patterns. In previous studies, see e.g. Scheidt and Caers [2009], Scheidt et al. [2011], variables such as the total oil production and water rates have been used as dissimilarity measures to assess the dynamical responses of different reservoir realizations. Despite that these variables are less coarse than the NPV, the production rates only generate local information of the flow behavior around the wells, and they have comparable restrictions as the dissimilarity measures based on NPV. The first attempt to use flow-based indicators as dissimilarity measure can be found in Yeh et al. [2014], where fingerprints are used to screen and cluster reservoir realizations with similar flow behavior. In this chapter, we propose the use of the full reservoir flow patterns as the dissimilarity measure between reservoir realizations. Reservoir flow patterns are the numerical solutions of the reservoir model in Eqs. (2.10)-(2.11), which are typically large-scale. The large dimensionality of these structures would make them unsuitable to serve as a dissimilarity measure for performing model discrimination and clustering, and therefore reduced-order representations are necessary. In several studies, see, e.g., Cardoso et al. [2009], Krogstad [2011], Markovinovic and Jansen [2006], the singular value decomposition (SVD) and POD model order reduction techniques have been applied to derive low-dimensional representations of flow variables, while in Yeh et al. [2014] the SVD has been used to represent the fingerprints through a reduced set of basis functions. However, the SVD approach has some limitations. As the reservoir flow patterns are stacked in vectors, the natural spatial-temporal structure of the reservoir is lost. This may have serious implications when characterizing flow profiles in low-dimensional spaces, as some information related to the spatial correlations is lost during the vectorization scheme, see, e.g., Insuasty et al. [2015b]. In this chapter, we develop a tensor approach for efficient storage of reservoir flow patterns in a multidimensional array. This creates a clear separation of the spatial, temporal and flow variables coordinates, and allows for reduced-order representations using basis functions in each of the separate coordinates, thereby appropriately maintaining spatial correlation structures. With an additional extension of the tensor coordinates, it will even allow for describing the flow characteristics of an ensemble of models. The corresponding reduced-order representations will be analyzed for their suitability to calculate distance measures between models, and for subsequent distance visualization and model clustering.

64 3.2 Flow-based dissimilarity measures Flow-based dissimilarity measures Introduction In water-flooding, the temporal evolution of the oil saturation (and in particular the oil-water front) provides sensible information for well placement and for the design of schedules for the well controls in order to optimize production. Hence, the reservoir flow patterns are the variables with physical interpretation that best describe the dynamic properties of the hydrocarbon reservoir, and we can conceptually state that two reservoir realizations are similar with respect to their dynamical performance if for a particular operational strategy, the generated reservoir flow profiles are similar. The variable s(x, t, u) will be used in this paper to represent the flow-related variable, with a spatial coordinate x R 2, time t R, and operational strategy u. In most cases s will correspond to the oil saturation in each (spatial) grid block, although other variables (e.g. pressure, time-of-flight, drainage time) could be included too. They are the solutions of the underlying model s multiphase flow equations through their corresponding PDE s, and the result of a particularly chosen operational strategy of water injection and control valve settings, reflected by the variable u, see, e.g., Jansen [2013]. In this section, we elaborate on the concept of model distances based on reservoir flow patterns Dissimilarity measures and distance functions When quantifying flow-based dissimilarities between reservoir models, one should consider the use of distance functions. A distance function defines the separation between two elements in a set (the set of reservoir flow responses) and it induces a metric space, where the distance between two different reservoir models is an indicator of their dissimilarity in the dynamical response. There are many functions to compute the distance between two objects: the Euclidean distance, standardized Euclidean, Chebyshev distances, and many more. In Suzuki and Caers [2008], the Hausdorff distance has been used to measure the dissimilarity of geometry for reservoir realizations, and in Park and Caers [2007] the connectivity distances has been computed based on streamlines. If s 1 and s 2 are the flow-related variables corresponding to two different models, a natural dissimilarity measure to consider is a quadratic distance measure: d(s 1, s 2 ) = K I k=1 i=1 j=1 J s 1 (x ij, t k, u) s 2 (x ij, t k, u) 2 (3.1) where K is the total number of time steps; I, J are the number of grid cells in each spatial dimension, and the two models are operated with the same operational strategy u(t k ). For brevity of notation we will often discard the dependency of s(x, t k, u) on u and simplify the notation to s(x, t k ) whenever there is no risk of confusion. The

65 44 Flow-based dissimilarity measures underlying spatial domain is assumed to be rectangular with Cartesian grid. The temporal evolution of the flow variables s(x, t k ) over all grid cells is a collection of high-dimensional state variables and generally requires the use of huge computational resources for storage, function evaluations, the evaluation of distances as in (3.1) and its subsequent use for visualization and model clustering. In the next section, a method for the low-dimensional representation of s(x, t k ) is described Low dimensional representations and flow-based distances through SVD Compact representations of s(x, t k ) are important for an efficient and fast numerical calculation of distance measures, see e.g. Yeh et al. [2014]. A typical way to construct lower dimensional representations of s(x, t k ) is obtained by utilizing a basis function expansion: s(x, t k ) = ˆR i=1 σ i (t k )ϕ i (x), (3.2) where the basis functions ϕ i (x), for i = 1,, ˆR can be selected to be the most informative spatial patterns in the flow response. If ˆR N, where N = I J is the number of grid cells, we say that the reservoir flow pattern s(x, t k ) is characterized in a low-dimensional space by the coefficients σ i (t k ) and by the basis functions ϕ i (x), for i = 1,, ˆR. The classical technique for obtaining this representation is through principle component analysis (PCA) and the use of singular value decompositions (SVD) (Golub and Van Loan [2012]). To this end the dynamic variables in the grid are represented as (I J) 1 vectors, denoted as x k, with elements s(x ij, t k ) at a particular time moment t k. A number K of these snapshot vectors x k is collected at time instants t 1,, t K, where K may be less than or equal to the total number of simulation time steps. With N = I J, this results in a N K matrix of data points which is decomposed using SVD through: X = [ x 1 x 2 x K ], (3.3) X = ΦΣΨ = R σ r ϕ r ψr = r=1 R σ r ϕ r ψ r, (3.4) where Φ and Ψ are N N and K K orthogonal matrices containing the left and right singular (column) vectors ϕ r and ψ r, Σ is an N K rectangular diagonal matrix that has the ordered singular values σ 1 σ 2 σ R 0 on its main diagonal, R is the rank of X, and denotes the tensor or outer product over a vector space. Usually, R K N in a typical reservoir simulation application. The last equality in (3.4) indicates that X can be decomposed as the sum of R rank-one matrices Θ r = ϕ r ψ r. In particular, every individual snapshot vector x k can be written as: r=1

66 3.2 Flow-based dissimilarity measures 45 R R x k = σ r ϕ r ψr e k = αrϕ k r, (3.5) r=1 r=1 where e k is the kth standard unit vector in R K and αr k := σ r ψr e k is a real-valued coefficient. The R left singular vectors ϕ r, r = 1,..., ˆR then characterize the spatial correlations of the original snapshot matrix X, ordered in decreasing relevance, and allows a low-dimensional approximation ˆx k = ˆR r=1 αk rϕ r, with ˆR < R, of the reservoir flow patterns and fingerprints in terms of the coefficients αr. k Using (3.5), the Euclidean distance between the ith and jth snapshots x i, x j is: d ij = x i x j = ( R ) 2 (αr i αr)ϕ j r = R (αr i αr) j 2. (3.6) r=1 r=1 Now, let us consider the low-rank approximation X = [ ˆx 1 ˆx 2 ˆx K ] of X, which can be obtained by decomposing (3.4) as X = X + X = ˆR r=1 σ r ϕ r ψ r } {{ } X + R σ r ϕ r ψ r, (3.7) r= ˆR+1 }{{} X where ˆR < R is the approximation order and where X = X X is the approximation error. For any ˆR < R, the Frobenius norm of the error X X F = is minimal over all rank ˆR approximations of X. The approximate representations {ˆx k } k=1, K of the reservoir flow pattern can now be used as a basis for measuring dissimilarities between models, where the appropriate calculations can be performed on the basis of the coefficients α k r for r = 1, ˆR and k = 1, K. Let us consider the low-dimensional characterization of the flow patterns in terms of the coefficients α i r, α j r for r = 1,..., ˆR, then the approximated dissimilarity is: d ij = ˆR r=1 r> ˆR σ 2 r (α i r α j r) 2 (3.8) where d ij is the (i, j)th element of a matrix D R K K of all approximate distances.

67 46 Flow-based dissimilarity measures Discussion The SVD-based approach presented in section 3.2 has been adopted in industrial practice, see e.g. Yeh et al. [2014], but it has some limitations. Through the vectorized form in which flow variables are stored, the spatial-temporal structure of the reservoir is lost. This may have serious implications when characterizing flow profiles in low-dimensional spaces. When SVD is applied to a snapshot matrix X, the sets of orthonormal basis vectors {ϕ r } ˆR r=1, {ψ r } ˆR r=1, average the energy of solutions in time, and by definition, do not discriminate among spatial coordinates. This temporal averaging of the energy causes a loss of information for some of the relevant features of spatial coordination in the data, see e.g. Insuasty et al. [2015b]. For linear systems like single-phase flow problems, the spatial correlations are invariant in time and can be characterized analytically using concepts from system theory such as controllability and observability, see e.g. Van Doren et al. [2013]. However, the nonlinearities induced by the multi-phase character of the problems may define time-variant correlations of the states, and the correlation between time and specific spatial direction is ignored by vectorizing the flow variables, in which case it can be attractive to clearly separate all spatial and temporal coordinates to maintain their own independent role when constructing approximations. In the next section, a methodology that overcomes the limitations of SVD methods for flow characterization is presented. 3.3 Spatial-temporal tensor methods for flow-based dissimilarity measures Introduction In this section, we develop a multidimensional approach to understand the dynamical similarities between reservoir models, which is based on tensor representations and decompositions of flow related variables. In addition, we incorporate this approach into a workflow for clustering of models with similar dynamical performance Tensor representations of reservoir flow patterns In the previous section, we have constructed vectorized representations of the flow variables (I J 1 snapshot vectors). Alternatively, if the spatial grid has a Cartesian structure, one can collect K snapshot matrices X k of size I J, and represent this data object in a three-dimensional array S of size I J K. That is, the reservoir flow data is represented as a multi-array S R I J K. Such a multidimensional array is called a tensor and can be viewed as the natural generalization of vectors and matrices to higher dimensional objects. For a 2D saturation field that evolves over time, a three-dimensional array is schematically depicted in Fig.(3.1).

68 3.3 Spatial-temporal tensor methods for flow-based dissimilarity measures 47 Figure 3.1: Schematic of the tensor representation of a reservoir flow pattern. Axes represent spatial-temporal coordinates. Color-scale corresponds to oil saturation A key advantage of multidimensional data objects is that they keep the spatial structure of the Cartesian grid intact. A disadvantage of the use of tensors is that their algebraic properties are more complicated and that numerical tools for tensor operations are less developed. In general, tensors are multilinear generalizations of algebraic objects such as vectors and matrices, and there exist suitable extensions of concepts such as decompositions, basis functions and spectral expansions to the multilinear case. In the next subsection, we describe the basic concept of tensor decompositions as an extension to the concept of matrix decomposition Tensor approximation of reservoir flow patterns Let us illustrate the concept of signal approximation and compression of reservoir flow patterns through tensor decompositions. In the framework of multidimensional (tensor) approximations, the sets of orthonormal basis functions {ϕ i } I i=1, {ψ j} J j=1 represent the most relevant spatial correlations independently for each spatial coordinate. The coordinate independence can be exploited to approximate flow patterns which have a richer variability in a certain coordinate, as it would be the case for flow patterns in channelized reservoirs. We consider a 2 facies, 2D oil reservoir with a square geometry of length L = 3000m, one layer of 10m thick. The numerical model of one realization has 3600 grid blocks of size 50m 50m. A description of the physical parameters, wells configuration,

69 48 Flow-based dissimilarity measures Figure 3.2: Oil-water front with tensor approximations. Approximation 1 has modal rank (20, 20, 5). Approximation 2 has modal rank (10, 10, 5). Colors represent oil saturation. and a link to the data files can be found in Vo and Durlofsky [2015]. The sequential solvers of MRST, see e.g. Lie et al. [2012], have been used to solve the pressure and saturation equations and the production has been simulated for a period of 15 years, time step of 5 days. There are 4 water injectors, and each of them injects at a rate of 600m 3 /day, and the producers operate at 150bar. We collect K = 1095 time steps for the temporal evolution of the oil saturation. Then, we construct a 3D tensor S of size , where the x, y and temporal dimensions correspond to the first, second and third tensor coordinate accordingly. Hence, the tensor S can be decomposed as in Eq. (5.17): S = σ ijk ϕ i ψ j χ k. (3.9) i=1 j=1 k=1 The reservoir flow pattern in tensor S is described by I = 60 basis functions of size 60 1 in the x coordinate, J = 60 basis functions of size 60 1 for the y coordinate, and K = 1095 basis functions of size for the temporal dimension. We compute low-rank approximations Ŝ of the original flow pattern by truncating the sums in Eq. (6.2). The purpose of this example is to study the effect of decreasing the

70 3.3 Spatial-temporal tensor methods for flow-based dissimilarity measures 49 number of spatial basis functions for the approximation, and therefore the number of temporal basis functions are fixed to {χ k } K=5 k=1. For the first approximation, we select basis {ϕ i }Î=20 i=1 for the x coordinate and basis {ψj}ĵ=20 j=1 for the y coordinate, leading to a modal rank approximation of (20, 20, 5). For the second approximation, we select {ϕ i }Î=10 i=1 basis for x and {ψ j}ĵ=10 j=1 basis for y, leading to a modal rank approximation of (10, 10, 5). Time snapshots of the reservoir simulation and the approximations are depicted in Fig. (3.2). Figure 3.3: Blue-Left axis: Relative proximity ν(r). Green-Right axis: Size in memory of the modal rank approximation (r, r, 5) as function of r. From Fig. 3.2, it is clear that decreasing the number of spatial basis functions would affect the quality of the approximations. This can be quantified by using (2.64) to derive the relative ( proximity ) ν of the approximations Ŝ with respect to the original tensor S: ν = 1 S Ŝ F S F 100. Here, we consider modal rank approximations of the type (r, r, 5), for r = 1,, 60, and the relative proximity as a function of r is presented in Fig For the flow patterns depicted in Fig. 3.2, it is observed that the approximation (10, 10, 5) preserves almost 85% of the features of Ŝ, while the approximation (20, 20, 5) preserves more than 90%. When fixing K = 5, only 0.46% of the total number of basis function for the temporal domain is used, while achieving a maximum proximity of 94%. For this example, that fact indicates that the temporal dynamics can be explained with a very small amount of the information contained in S. The amount of information required to construct an approximation can be quantified by summing the size in memory of the constitutive elements in (2.63): {ϕ i }Îi=1, {ψ j}ĵj=1, {χ k} K=5 k=1, and the corresponding core tensor Σ. Similarly, we consider modal rank approximations of the type (r, r, 5), for r = 1,, 60, and the information is presented in Fig The size in memory of the tensor S is 30.08MB, while the approximations (20, 20, 5) and (10, 10, 5) of Fig. 3.2 have a size of 79.38KB and 57.78KB respectively. These findings indicate that the reservoir flow pattern in S can be approximated using only 0.25% of its original

71 50 Flow-based dissimilarity measures Figure 3.4: Schematic interpretation of a 4D tensor of reservoir flow patterns. information, while achieving relative proximities higher than 90%. This experiment suggests that more than 99% of the information contained in S is redundant for the purpose of flow characterization D Tensors: an approach for handling multiple realizations In the multilinear framework, it is possible to define an additional coordinate, where an index that links the dynamical behavior (the reservoir flow pattern) to its corresponding model is assigned to every realization. In this subsection, we restrict our attention to 2D cartesian grids, without losing generality for 3D geometries. For the case where we have an ensemble of R realizations and their corresponding flow patterns, the full data set is described by two spatial coordinates, the temporal coordinate and a coordinate for the realizations, i.e. a 4D tensor S of size I J K R, with I, J the dimension of the spatial coordinates x and y, K the dimension of the temporal coordinate, i.e., the number of time steps, and R the number of reservoir models, which constitutes the size of the ensemble to be characterized. A schematic representation of such a data structure is depicted in Fig In analogy to the Eq. (5.17), the 4D tensor S has a Tucker decomposition of the form: S = I J K i=1 j=1 k=1 r=1 R σ ijkr ϕ i ψ j ω k χ r, (3.10) where the orthonormal basis vectors {ϕ i 1 i I}, {ψ j 1 j J} span the spatial coordinates, {ω k 1 k K} spans the temporal space and {χ k 1 k R} spans the model space. The reservoir flow patterns of the mth realization X m is then represented as a tensor of dimension I J K which is approximated similar as in (2.66), and results in the approximate sample X m of the flow patterns defined by the order-3 tensor

72 3.3 Spatial-temporal tensor methods for flow-based dissimilarity measures 51 X m = Ŝ(,,, e m) = = Î Ĵ K R i=1 j=1 k=1 r=1 K Î Ĵ i=1 j=1 k=1 σ ijkr χ r, e m ϕ i ψ j ω k where e m is the mth standard unit vector in R R and where αijk m is a real-valued coefficient in the expansion (6.4). α m ijkϕ i ψ j ω k, (3.11) := R r=1 σ ijkr χ r, e m This expansion shows explicitly the way how the information is distributed in the decomposition. Clearly, the tensor ϕ i ψ j ω k contains the spatial-temporal correlations that are shared by all the set of reservoir flow patterns in the ensemble. What makes a reservoir flow pattern X m distinct from others is the selection of the mth element of the basis functions for the model coordinate χ r, e m and subsequently the coefficients α m ijk. As it was indicated previously for the 3D case, the coefficient αijk m is a linear combination of the m-th element of all the basis functions in the set {χ r } R r=1, i.e., αijk m = R r=1 σ ijkr χ (m) r, and the information that characterizes the dynamical properties of the realizations are embedded into the elements of core tensor σ ijkr and the set of basis functions for the model coordinate {χ r } R r=1. This analysis creates the foundations for the definition of low-dimensional representations of the flow profiles in the next subsection Flow-based dissimilarity measures in low-dimensional tensor representations In order to be able to calculate a dissimilarity measure between two models on the basis of low-dimensional representations, we require the tensor representation of the flow patterns for both models to be expanded with the same basis functions, as in Eq. (6.4). Therefore we construct the 4D tensor representation described in Section 3.3.4, and we introduce a metric space by defining a distance function between two reservoir flow patterns X p and X q of the realizations p, q as: d pq = X p X q F I J K R ] = σ ijkr [ χ r, e p χ r, e q ϕ i ψ j ω k i=1 j=1 k=1 r=1 F I J K R = σ ijkr χ r, e p e q ϕ i ψ j ω k, (3.12) F i=1 j=1 k=1 r=1

73 52 Flow-based dissimilarity measures where e p, e q are the pth and qth standard unit vectors in R R. Let us define the realvalued coefficient δ ijk = R r=1 σ ijkr χ r, e p e q, which is an element of a tensor D of dimensions I J K. Hence, (3.12) can be written as d pq = I J i=1 j=1 k=1 K δ ijk ϕ i ψ j ω k = D F Φ 2 Ψ 2 Ω 2, (3.13) F where Φ,Ψ and Ω are column matrices composed by the basis functions {ϕ i 1 i I}, {ψ j 1 j J} and {ω k 1 k K}. Due to the orthonormality of the columns of Φ,Ψ and Ω we obtain: d pq = D F = I J K δ ijk 2. (3.14) i=1 j=1 k=1 The expression in (3.14) suggests that the dissimilarity between two reservoir realizations can be approximated by truncation, which corresponds to the expression Î Ĵ d pq = K i=1 j=1 k=1 δ ijk 2. (3.15) In addition, the scalar δ ijk can be expressed in terms of the coefficients αijk m in (6.4), δ ijk = α p ijk αq ijk, and therefore the tensor-based approximation of the flow-based distance between the realizations p and q is defined as Î Ĵ d pq = K i=1 j=1 k=1 α p ijk αq ijk 2. If one stores the set of coefficients α m ijk as elements of a tensor A m of dimension Î Ĵ K, then Î Ĵ d pq = K i=1 j=1 k=1 α p ijk αq ijk 2 = Ap A q F, (3.16) where d pq is the pq th element of a distance matrix D. The approximation error of computing the dissimilarity measure using the approximations in (6.4), is bounded, see, e.g., De Lathauwer et al. [2000a].

74 3.4 A tensor-based workflow for model clustering using flow measures 53 The approximation of the distance in (3.16) can be computed based on the tensors A p and A q, and they can be seen as compact representations of the pth and qth reservoir flow patterns. The set of tensors A = {A 1, A 2,, A R } is composed by low-dimensional representations of the reservoir flow patterns for an ensemble of R realizations, and they are used for further steps in flow characterization such as distance visualization and model clustering Discussion With the spatial-temporal methodology it is possible to identify the tensor coordinates with richer information content, and it introduces flexibility and extra accuracy when representing flow patterns in low-dimensional spaces and calculating dissimilarity measures. Flow-based dissimilarity measures allow the classification of the different types of flow behavior in an ensemble, and the application of the methods described in this section are useful for model clustering, where the computational complexity limits the classification of full reservoir flow patterns. In the next sections, we apply the concept of flow-based dissimilarity measures in low-dimensional spaces to the flow classification of reservoir models using the tensor approach. 3.4 A tensor-based workflow for model clustering using flow measures Introduction In reservoir engineering, multiple realizations are used to account for the uncertainty of the rock properties of subsurface, and the industrial practice indicates that despite the fact that realizations look different from the geological perspective, some of them may have similar dynamical performance. In flow classification, we aim to find sets of realizations that share a similar dynamical performance with respect to the spatial-temporal evolution of their corresponding reservoir flow patterns. For that, it is required to compute dissimilarity measures between related flow patterns, a method for visualizing these dissimilarities and a clustering technique to group the models with similar dynamical properties. When using a flow-based dissimilarity measure for model clustering, these steps are constrained by the dimensionality of the data set to be analyzed, and the representation of the reservoir flow patterns in low-dimensional spaces are used for the efficient classification of multiple reservoir realizations. The flow-based approach for dissimilarity measures was introduced in section 3.3. Here we provide the theoretical foundation for the workflow developed in this paper. In subsection we describe a tensor-based clustering algorithm, and in subsection we describe the method for visualizing distances based on the distance matrix D.

75 54 Flow-based dissimilarity measures k-means tensor clustering When analyzing data sets, analysts aim to extract patterns, object classification and data ordering. Thereby, k-means clustering finds groups of data which are similar to one another, partitioning a set of objects into clusters. Let us consider the data objects described in section 3.3.5, where the tensor data set A = {A 1, A 2,, A R } is composed by the low-dimensional representations of the flow patterns. In this section, we aim for a partition of the data set A into a set of K c clusters C = {c 1, c 2,, c Kc } with corresponding centroid µ k of dimension Î Ĵ K, the same size of the elements in the set A, where c k A for k = 1, K c, such that the variance within each cluster is minimized, see e.g. Jegelka et al. [2009]. This operation can be formulated as: arg min C K c A i µ k 2, subject to: µ F k = 1 N k k=1 i c k j c k A j, (3.17) where N k is the number of elements (size) of the cluster c k and j c k A j indicates the element-wise sum of the tensors A j which have been assigned to the cluster c k. The k-means algorithm has NP-hard complexity (Aloise et al. [2009]), which can be relaxed using heuristic algorithms like the Lloyd s algorithm, see e.g. Lloyd [1982]. The algorithm has two basic steps: 1) The assignment of every tensor object in A to the closest cluster centroid, and 2) the re-computation of the centroids using the current cluster membership: Initialize cluster centroids µ 1, µ 2,, µ Kc randomly. Repeat until convergence: 1. Label assignment step: Assign each data point to the nearest centroid. For j = 1,, R and k = 1,, K c perform: 2. Clusters update: Update the set C. l j = arg min k Aj µ k 2 F. (3.18) 3. Centroids update: Compute the average of the cluster elements. µ k = 1 N k i c k A i. (3.19) The selection of the K c is a user choice, however there are more systematic methods to determine the initial guess for the number of clusters, see Ketchen and Shook [1996]. When working with large-scale data sets, it is required to account for the scalability and the computational complexity of the algorithms for data analysis. Lloyd s algorithm has linear computational complexity O(it K c R n), where it is the number of iterations needed to converge and n is the size of the objects to be clustered

76 3.4 A tensor-based workflow for model clustering using flow measures 55 respectively. From the complexity point of view, the application of the k means tensor clustering algorithm is constrained by n, i.e. the dimensionality of the objects to be clustered. Particularly, the fact that the reservoir flow patterns are large scale data structures (n 10 6 ) poses a challenge for the state-of-the-art clustering algorithms, and limits the use of k-means for flow-based classification. Hence, the use of low-dimensional representations of flow patterns, such as the elements of the set A, makes the implementation of clustering algorithms feasible in practice using current computational resources Visualization of dissimilarities For the visualization of dissimilarities between a set of reservoir flow patterns, we determine their coordinates in a metric space using Multidimensional scaling (MDS). For a detailed description, we refer to Borg and Groenen [2005]. For the application of MDS in uncertainty quantification, we refer to Caers et al. [2010], Scheidt et al. [2009], Suzuki et al. [2008]. MDS uses the SVD to determine a low-order set of dimensionless directions in which the relative distances between the objects can be efficiently represented. In particular when considering just two or three of the most relevant directions it is possible to represent the distances between the objects graphically. Following (3.16), and given a distance matrix D of a set of R low-dimensional representations of reservoir flow patterns, where d ij is the ij-th element of D, MDS finds the coordinate vectors x 1, x 2,..., x R R R in a metric space, such that x i x j d ij. Let x i = [x (1) i, x (2) i,, x (R) i ] and x j = [x (1) j, x (2) j,, x (R) j ] be the coordinate vectors for the realizations i and j correspondingly, then the squared distance between the points is given by d 2 ij = R (x (r) i r=1 x (r) j ) 2 = x i x i + x j x j 2x i x j, (3.20) which can be expressed using matrix notation as = Y1 + 1Y 2XX. Here, d 2 ij is the ij th element of the matrix, Y = [x 1 x 1,, x R x R] R R, 1 R R is the vector with all elements equal to 1, and X R R R is the matrix with the ith row equal to x i. Here we solve the reverse problem, where given (from the distance matrix D), we determine the coordinate matrix X corresponding to the location of the models in a metric space. For that, let us define the centering matrix of rank L as C L = I L 1 L 11, where I L is the identity matrix of size L. The centering matrix C L when multiplied with a vector subtracts the average value of the vector to all of its components. A pre and post multiplication of by C L leads to: C L C L = C L Y ( 1 C L ) + ( CL 1 ) Y C L 2C L XX C L (3.21) In Eq.(3.21), the terms ( 1 C L ) = 0 and ( C L 1 ) = 0, because the average of the

77 56 Flow-based dissimilarity measures components of 1 is 1. The term C L XX C L = XX under the assumption of a centered X. Then, XX = 1 2 C L C L. An eigenvalue decomposition of XX leads to XX = 1 2 C L (2) C L = UΣU = UΣ 1/2 Σ 1/2 U, (3.22) then, the coordinate matrix is given by X = UΣ 1/2. To generate the 3D MDS plot, we select the first 3 columns of X, and every row corresponds to the location of a model in a 3D cartesian map A workflow for model clustering using flow measures In this subsection we use the multilinear algebra methods described in this paper to find clusters of models with similar dynamical properties. The developed methodology uses the concept of flow-based dissimilarity measures, computed in low-dimensional spaces to determine the dynamical similarities between reservoir models, by exploiting the tensor structure of the reservoir flow patterns. The purpose of this workflow is to estimate the closeness between two or multiple realizations with respect to a performance indicator relevant to the CLRM framework. The inputs for the workflow are: R: The number of realizations. X i : Time snapshots of the reservoir flow patterns (i = 1,, R). K c : The number of clusters. A predefined production strategy u(t). The procedure is described as follows: 1. Reservoir simulation: Simulate the flow patterns for the set of R realizations using u(t). 2. Tensor formulation: Store the reservoir flow patterns of all the realizations in a tensor S as described in section Decomposition: Compute the tensor decomposition of S as in Eq. (3.10), using the algorithms described in section Low-dimensional characterization: Construct the low-dimensional representation of the flow profiles A = {A 1, A 2,, A R }, as described in section Dissimilarity: Compute the distances described in Eq. (3.16). 6. Clustering: Group the data set A into clusters as described in section

78 3.5 Application case Visualization: Construct an MDS map to visualize clusters as described in section The output of the workflow is the set of K c clusters C = {c 1, c 2,, c Kc }, which groups the types of dynamical responses of R reservoir realizations. This classification can be further used to create flow-relevant ensembles, where few reservoir models are selected to capture the most relevant dynamical responses of the original set of realizations. Figure 3.5: Well configuration and samples of permeability fields from the ensemble. Color scale in mdarcy 3.5 Application case In this section, the workflow for model clustering presented in section 3.4 is applied to a set of channelized reservoirs, and we analyze the performance of the spatialtemporal approach using flow-based dissimilarity measures. Channelized reservoirs present a challenge for field development plans, because moderate changes in well configurations may lead to very high variations in the resulting reservoir flow patterns. Let us consider an ensemble of R = 100, 3D reservoir models with a geological structure consisting on a network of fossilized meandering channels of high permeability. The data set has been uploaded to the 4TU.Datacentrum repository and can be accessed by external users, see e.g. Jansen et al. [2014] for the physical parameters of the models. The reservoir size is 480m 480m 28m with 7 geological layers, and it is composed by grid blocks 8m 8m 4m in size. We have used a rectangular-shaped geometry instead of the egg-shaped reservoir described origi-

79 58 Flow-based dissimilarity measures nally in Jansen et al. [2014]. The well configuration is composed of 8 injectors and 4 producers. A view of some geological realizations is depicted in Fig Figure 3.6: Approximation error e r (Î, Ĵ) = S Ŝ(Î,Ĵ) F S F Generation of the reservoir flow patterns We have simulated the reservoir flow patterns of the R = 100 realizations, which correspond to the spatial-temporal evolution of the oil saturation. The sequential solvers of MRST, see e.g. Lie et al. [2012], have been used to solve the pressure and saturation equations, and the production has been simulated for a period of 10 years with a time step of 30 days, i.e. K = 122 time steps. The water injection rates are fixed at 79.5m 3 /day for all the injectors and the bottom-hole pressures are fixed at 395bar for all the producers Low-dimensional tensor representation of the reservoir flow patterns The data structure that contains the reservoir flow patterns for the ensemble can be stored in a 5D tensor of size I J Z K R with I = 60 the dimension of the x coordinate, J = 60 the dimension of the y coordinate, Z = 7 layers, K = 122 time steps and R = 100 realizations. The tensor S is decomposed similarly to the

80 3.5 Application case 59 Figure 3.7: Snapshots of oil saturation for model 57 (layer 3) with Î = 10, Ĵ = 10, Ẑ = 2, ˆK = 2 and ˆR = 100. Top: Reservoir simulation. Bottom: Tensor approximation. decomposition in (3.10), while augmenting a coordinate for geological layers. Hence, the reservoir flow patterns corresponding to the mth realization can be described as: X m = I J Z i=1 j=1 z=1 k=1 K αijzkϕ m i ψ j ν z ω k, (3.23) where {ν z } Z z=1 are the set of basis functions for the layers coordinate and α m ijzk := R r=1 σ ijzkr χ r, e m. For the low dimensional approximation of S, we truncate the number of the basis functions in every coordinate such that the approximation error described in (2.64) is relatively small. We have set the number of basis functions for the layers coordinate to be Ẑ = 2, and for the temporal coordinate to be ˆK = 2. From the expression in (6.4), it is inferred that the number of basis functions for the model coordinate does not affect the number of parameters α ijzk required to describe a flow pattern, and thus we select ˆR = 100 basis functions for the model coordinate. In order to choose an adequate number of spatial basis functions for the x and y coordinates, we perform an error analysis using the approximation error defined in (2.64). In Fig. 3.6, the approximation error as a function of the number of basis functions used for the approximation Ŝ is presented. Using Fig. 3.6, we can perform a tradeoff between accuracy and the amount of information required for the approximation. The truncation criterion is defined by the user, and we accept tensor approximations

81 60 Flow-based dissimilarity measures with relative errors lower than 20% with respect to the original tensor S. By choosing a truncation Î = 10 and Ĵ = 10 we achieve a relative error of 17.6% with respect to the flow patterns from the full simulation. Fig 3.6 displays an almost symmetric shape, and shows that the approximation error tends to zero as we increase the number of basis functions in both coordinates. Snapshots of the approximation for one of the realizations are depicted in Fig Despite that the oil/water front is not as sharp as in the reservoir simulation, the tensor approximation is able to capture the relevant flow patterns as time evolves. These approximations allow the characterization of the reservoir flow patterns in low-dimensional spaces. As was described in the previous section, the reservoir flow pattern of the mth realization X m (of size I J Z K, i.e grid-block oil saturations) is characterized in a low-dimensional space by the tensor A m of size Î Ĵ Ẑ K composed by the set of coefficients α ijzk, i.e. 400 coefficients. This low-dimensional characterization represents a reduction of 99.9% of the amount of information necessary for further classification analysis. Figure 3.8: MDS plot. Color represent flow-based clusters. Numbers are assigned to all the realizations.

82 3.5 Application case 61 Figure 3.9: MDS plot. Left: Color represents NPV (USD). Right: Color represents total oil production (stb). Numbers are assigned to all the realizations Model clustering and visualization The low-dimensional tensor representation in (3.23) and the methods described in section are used for model clustering. In the previous section, we have derived a set of tensors A = {A 1, A 2,, A R } which are low-dimensional representations of the reservoir flow patterns for an ensemble of R realizations. Here, we use them for constructing model clusters with similar reservoir flow patterns. The k-means clustering algorithm described in section has been used to classify the set A. The algorithm is provided with the number of clusters K c = 7, which have been selected by visual inspection of the MDS plots depicted in Fig A predefined initial condition for all the centroids µ 0 = 0.1 E, where E is a tensor of size with all the elements equal to 1. To visualize the clusters, the MDS map is constructed using a distance matrix D based on the approximated dissimilarity function defined in (3.16). In the MDS plot, every dot corresponds to a low-dimensional representation of the reservoir flow patterns for one realization, and the colors indicate clusters. The clusters depicted in the MDS plot are visually separated and the plot has a tetrahedron shape. Big clusters are found in the corners of the tetrahedron, indicating 4 predominant and different types of flow patterns. Typical flow patterns of these clusters are depicted in Figs To describe the observations, let us analyze the type of reservoir flow patterns classified in some of the clusters. In Fig. 3.10, the flow patterns corresponding to the realizations 9 and 64 indicate a large connectivity between the injectors 2, 4 and 7 with the producers 2 and 3 resulting in flow patterns with an elongated shape in

83 62 Flow-based dissimilarity measures Figure 3.10: Snapshots of oil saturation (top layer) of sample models from Cluster c 4. Figure 3.11: Snapshots of oil saturation (top layer) of sample models from Cluster c 2

84 3.5 Application case 63 Figure 3.12: Snapshots of oil saturation (top layer) of sample models from Cluster c 1 Figure 3.13: Snapshots of oil saturation (top layer) of sample models from Cluster c 3

85 64 Flow-based dissimilarity measures the y coordinate. In Fig. 3.11, the flow patterns of the realizations 3 and 38 present a large connectivity between injectors 1, 2, 3 and 4 with producers 1, 2 and 3 resulting in a rounded flow pattern in the center of the reservoir. In Fig. 3.12, the flow patterns indicate a large connectivity between injectors 1, 2 and 3 with producer 1, while the flow patterns corresponding to the cluster in Fig do not exhibit such connectivity. The results depicted in Figs confirm the effectivity of the spatial-temporal workflow for model clustering using flow-based dissimilarity measures NPV and oil production in the clusters Previously, we have discussed the fact that models with similar outputs such as NPV or production rates might have very different reservoir flow patterns. In the context of model clustering with flow-based dissimilarity measures, it is expected that the models within a cluster share similar flow patterns. We anticipate that the NPVs and total oil productions are similar as well, based on the fact that models with similar flow patterns might have similar evolution of the oil saturation and pressure patterns, generating close water breakthrough times and rates at the production wells. For the application case considered in this section, we compute the undiscounted NPV as in Jansen et al. [2008], with r o = 55USD/stb the oil price, r wi = r wp = 2USD/stb the cost associated to water injection and production. The range of NPV for the ensemble is ω NP V := [ , ] USD, with a mean value of µ NP V = USD and standard deviation σ NP V = USD. The statistical properties (mean and standard deviation) of NPV and total oil production of the flow-based clusters are presented in Table 3.1. The averages of the NPV of clusters are distributed in the range ω npv, and all of the intra-cluster standard deviations are smaller than σ NP V, suggesting NPV distributions around the cluster averages with smaller variances. Similar results are found for the intra-cluster distribution of total oil, with an ensemble average of µ OIL = stb and standard deviation of σ OIL = stb. All of the intra-cluster standard deviations of total oil are smaller than σ OIL. The distribution of NPV and total oil for the clusters can be visualized in the MDS plots of Fig From visual inspection, patterns of similar NPV and total oil production can be detected for models which belong to the same cluster and are close in the flow patterns Input dependency of the reservoir flow patterns and dissimilarity measures One of the possible limitations of the workflow is the nonlinear dependency of the reservoir flow patterns on the type of production strategy. In this section, we compare the results of clustering using flow-based dissimilarity measures for different production strategies. The strategies that will be considered in this section consist on a fixed bottom-hole pressure of 395bar at the producers, and the injection rates described in Table 1.

86 3.5 Application case 65 Table 3.1: Cluster distribution of NPV and total oil. Cluster NPV (10 6 USD) Total Oil (10 3 stb) No. Nk: Cluster size µnpv σnpv µoil σoil Ensemble Table 3.2: Production strategies for the sensitivity analysis. Strategy Injection rates Deviation Base case r = 79.5m 3 /day 2 rodd = r 1, reven = r = 4m 3 /day 3 rodd = r + 1, reven = r 1 1 = 4m 3 /day 4 rodd = r 2, reven = r = 10m 3 /day 5 rodd = r + 2, reven = r 2 2 = 10m 3 /day

87 66 Flow-based dissimilarity measures In general, it would be beneficial to have a small sensitivity of the workflow to variations in the control input. In order to assess this sensitivity, we have applied the workflow for flow characterization with the control strategies described in Table 3.2, and have generated K c = 7 clusters for each strategy. Here, we define the closeness of the clusters with deviated control to the clusters of the base case, as the number of shared models among the clusters. This cluster similarity can be quantified as the percentage of the elements shared with the cluster in the base case. Let c base r and c n r be the r-th clusters for the base and the n-th production strategy. We define the percentage of similarity of the cluster c n r with respect to the base case c base r as: p r = card(cbase r c n r ) card(c base 100, (3.24) r ) where card( ) denotes the set cardinality, i.e. the number of elements of a set. In Fig. 3.14, this percentage of similarity is presented. Figure 3.14: Model clusters similarity with respect to clusters of the base case. The results in Fig indicate that for small deviations (strategies 1, 2), there is a good agreement of the clusters for deviated controls with the classification obtained by the base case, as most of the clusters match the base case with at least 68%, with the exception of the cluster 7 for the strategy 2. This is expected, as there are no large variations in the reservoir flow patterns for small variations in the controls. For large deviations in the control inputs (strategies 3, 4), the similarity of the clusters with respect to the base cases decreases, however, there it is still a good matching, and the generated clusters have a significant similarity with respect to the classification found in the base case.

88 3.6 Concluding remarks 67 The results indicate that the nonlinear dependency of the flow patterns on the control input is an inherent limitation of the workflow. However, this methodology is valid for small deviations around the production strategy. 3.6 Concluding remarks Some relevant advantages have been identified for the proposed methodology of model classification using flow-based dissimilarity measures and tensor representations: Firstly, the spatial structure of the reservoir is preserved, which allows the extraction of spatial correlations from the reservoir flow patterns. Secondly, the spatial correlations are not averaged in time, which is particularly useful for the flow characterization of nonlinear reservoir systems, where the spatial correlations of the reservoir variables are time-variant. Thirdly, a tensor-based representation provides the user with enough flexibility for handling multidimensional reservoir flow patterns and for performing a directional approximation of the data, i.e., keeping the directions where the dynamics have a higher variability. As a consequence, the tensor approximations can represent patterns in the full simulation using only 0.1% of the original information. Finally, a low-dimensional representation of the reservoir flow patterns allows the implementation of dissimilarity and clustering techniques for reservoir models. The workflow developed in this paper classifies flow patterns, which due to their size cannot be compared otherwise. One of the possible limitations of the workflow is its sensitivity to the choice of control input, and the results for clustering and visualization lead to the assessment of dynamical similarities for small deviations around the selected production strategy. The applications of this type of reservoir flow characterization can be found in defining flow-relevant ensembles for data assimilation, production optimization and the assessment of the value of information in CLRM.

89

90 4 CHAPTER Reduced-size ensembles in CLRM with flow-based dissimilarity measures 1 This chapter presents the application of the workflow to select representative model realizations from an ensemble, based on their flow similarities, to the production optimization and parameter estimation loops of CLRM. Previously, tensor techniques have been exploited to represent the reservoir flow patterns in low-dimensional spaces, which allowed the efficient evaluation of flow-based dissimilarity functions and the implementation of clustering algorithms. In the next sections, we elaborate on the use of clustering methods described in Chapter 3 to arrive at reduced computational complexity of robust optimization and parameter estimation. 4.1 The selection of representative realizations The application of robust methods for optimization, ensemble-based parameter estimation, and the assessment of the value of information for realistic cases is limited by the dynamical complexity of the reservoir models, see, e.g., Barros et al. [2015], Fonseca et al. [2014a], Van Essen et al. [2009]. In order to accelerate those procedures, several authors have developed methods for the selection of representative realizations: On the one side, Ballin et al. [1992] have approximated the cumulative density functions (CDFs) of a response variable (e.g. total oil production) in an ensemble of models by using few realizations and tracer simulations. Odai and Ogbe [2011] have approached the problem of ranking realizations for fluid simulation by comparing their volume of oil originally in place, the cumulative recovery, the average breakthrough times from streamline simulation and the geometrically averaged perme- 1 Part of the content of this chapter is also published or presented in E. Insuasty, P. M. J. Van den Hof, S. Weiland, and J. D. Jansen. Spatial-temporal tensor decompositions for characterizing control-relevant flow profiles in reservoir models. In SPE Reservoir Simulation Symposium., Houston, Texas,

91 70 Flow-relevant ensembles in CLRM ability. On the other side, Sarma et al. [2013] have used a min-max approach where representative models of an ensemble have been selected based on their similarities with respect to static measures such as oil production. The models are selected such that they match some production target percentiles (i.e. 10 th, 50 th and 90 th percentiles of oil in place) while maximizing the variability in their geological structures. Sharifi et al. [2014] have used the fast-marching method to approximate the pressure and front-propagation to rank large geocellular models and to rank them according to stock tank oil initially in place (STOIIP), while McLennan and Deutsch [2005a,b] and Li et al. [2012] have used static geological measures (statistical calculations of connectivities, original oil in place, average porosity, etc.) for the model classification. Ates et al. [2003] have used the volumetric sweep efficiency computed via streamline simulations and time of flight (TOF) to select representative models for target percentiles of production of a realistic Middle Eastern carbonate reservoir. Particularly for parameter estimation and history matching, Yeh et al. [2014] have incorporated the generation of reduced-size ensembles based on flow indicators (TOF and drainage time), called fingerprints, in a probabilistic history matching workflow, and Jimenez et al. [2015] have applied the fingerprint technique for history matching of a realistic-sized shallow marine fluvial system. In this Chapter, we present a method to create a reduced-size ensemble from a set of clusters of reservoir models, which is aligned with the methods described in Chapter 3. We have assumed that the ensemble of models is a sampled subset of the parameter space and captures appropriately the geological complexity of the subsurface, and the method described in this section is based on clusters of realizations of different sizes. As was described in section 3.4.4, the output of the workflow for flow characterization is a set of K c clusters C = {c 1, c 2,, c Kc }, with corresponding cluster sizes R 1, R 2,, R Kc respectively, where R = R 1 + +R Kc and usually R i R j for i, j = 1,, K c. Here, a cluster c i with a very large R i (number of realizations) means that there is a highly representative dynamical behavior present in the whole ensemble, which is shared by the elements of c i. In contrast, a cluster c j with a very small R j means that there is a very atypical dynamical behavior in the whole ensemble, and the realizations which are part of c j may represent model outliers. The idea of creating flow-relevant ensembles is based on the notion of selecting few but representative realizations from the cluster set C, such that the relevance of each cluster c i is represented in the flow-relevant reduced-size ensemble. For instance, we expect that clusters c i with larger numbers of elements contribute with more reservoir realizations to the creation of the flow-relevant reduced ensemble. In order to describe the method, let us assign the membership probability p i of belonging to the cluster c i as p i = Ri R. Here, p i is an indicator of the relative relevance of the cluster c i in the whole ensemble, and p 1 + p p Kc = 1. In order to create a flow-relevant reduced-size ensemble, we first define the size of the reduced ensemble R red R. Subsequently, for every cluster c i, only Ri red < R i realizations are selected randomly, such that R red = R1 red + + RK red c. The number of elements Ri red selected from the cluster c i is defined by:

92 4.2 Assessing the quality of the reduced-size ensembles 71 R red i = R red p i. (4.1) To exemplify the selection mechanism, let us consider an ensemble of R = 120 realizations, which has been clustered according to the workflow in Section 3.4 into three clusters C = {c 1, c 2, c 3 } with sizes R 1 = 60, R 2 = 40 and R 3 = 20. For instance, the membership probabilities associated with each of the clusters c 1, c 2, c 3 are p 1 = 0.5, p 2 = 0.33 and p 3 = 0.17 respectively. If the objective is to construct a flow-relevant ensemble of size R red = 24 realizations, then, according to Eq. (4.1), we select randomly R1 red = 12 realizations from cluster c 1, R2 red = 8 realizations from cluster c 2 and R3 red = 4 realizations from cluster c 3. This example is summarized in Fig Figure 4.1: Flow-relevant ensemble The circle chart in Fig. 4.1 shows the percentage of realizations selected randomly from the clusters c 1, c 2, c 3 in order to construct a flow-relevant ensemble of R red = 24, which corresponds to the 20% of the original ensemble size R = 120. In the next sections, we apply this selection mechanism and the workflow for model clustering presented in Chapter 3 to accelerate the parameter estimation and production optimization loops of the CLRM scheme. 4.2 Assessing the quality of the reduced-size ensembles Previously, we have described a weighted selection mechanism for realizations based on their cluster membership. In this section, we assess the quality of the reduced ensembles with respect to their ability to reconstruct CDFs of a target variable, see, e.g., Ballin et al. [1992] and Yeh et al. [2014].

93 72 Flow-relevant ensembles in CLRM Figure 4.2: Geometry and well configuration of the Brugge field The Brugge field In order to assess the quality, we test our methods with the Brugge field benchmark. The Brugge field was introduced at the SPE applied technology workshop on CLRM in Brugge in Since its introduction, it has been a popular benchmark to test novel methodologies for the optimization and data assimilation loops of the CLRM scheme, see e.g., Peters et al. [2013] and Chen et al. [2010]. The Brugge field consists of 9 layers of an E-W elongated dome with a boundary fault located at the northern edge, and one internal fault pointing to the northern edge. We refer to the work of Peters et al. [2010] and Lorentzen et al. [2009] for a detailed description of the petrophysical properties and initial conditions. The Cartesian grid of the Brugge field consist of I = 139, J = 48 and Z = 9 divisions in the x, y and z coordinates respectively. TNO (a Dutch applied research organization) has provided the users with 104 realizations of the petrophysical properties (permeability, porosity and net-to-gross ratio). We consider realization number 104 to be the truth model, and the remaining R = 103 realizations to be the ensemble that captures the geological uncertainty. The spatial grid consists of grid cells and the reservoir is operated with 10 peripheral injectors and 20 producers. One of the realizations and the well configuration of the field is depicted in Fig For parameter estimation, the Brugge field has been used by Luo [2015] to test an unified framework for ensemble data assimilation (Kalman and smoother filters). Kam et al. [2016] have used the benchmark to test a novel update methodology which incorporates tracer-surveillance data and distributed water arrival time information. Chang et al. [2015, 2016] have combined the facies parametrization with the adaptive Gaussian mixture filter for the data assimilation of a channelized version of the field. Other authors that have used the benchmark to test parameter estimation schemes are Backhouse et al. [2011], Mohamed et al. [2010], Thomas et al. [2011].

94 4.2 Assessing the quality of the reduced-size ensembles Flow-based model classification In order to cluster realizations with similar dynamical performance, we follow the methodology described in Section 3.4. The sequential solvers of MRST provided by Lie et al. [2012], have been used to solve the pressure and saturation equations for the water-flooding production stage. In this section, we use a fixed production strategy where each of the injector wells injects water at a rate of 4000stb/day, and the producer wells operate at a bottom-hole pressure of 1850psi. As was described in Chapter 3, the tensor formulation of the reservoir flow patterns naturally allows the inclusion of more than one flow variable for the analysis. In other words, instead of considering only the oil saturation we can also provide a combination of oil saturation and pressure data, which increases the information content used for the model discrimination step. In this section, we exploit the flexibility of the tensor formulation to add extra information in the form of flow variables. Let us consider two cases of reservoir flow patterns, which are described in 4.1: Case 1 includes the oil saturation s o, and Case 2 considers s o and pressure p simultaneously. The simulation time step t = 3 months is the same for all the cases. Flow patterns Simulation Time Variables Time samples Case 1 30 years s o K 1 = 120 Case 2 30 years s o and p K 2 = 120 Table 4.1: Characteristics of the flow patterns Tensor representation and approximation of the flow patterns Under the simulation conditions described in Table 3, K 1 and K 2 time samples have been collected for R = 103 model realizations. The resulting tensors which contain the flow patterns are described as follows: Case 1: the data structure that contains the reservoir flow patterns (oil saturation) are stored in a 5D tensor S 1 of size , where the coordinates correspond to information in the x, y, z, time and model coordinates respectively. Case 2: we have incorporated an extra coordinate to the tensor structure in order to handle the flow variables (oil saturation and pressure).the reservoir flow patterns are stored in a 6D tensor S 2 of size , where the coordinates correspond to information in the x, y, z, time, model and variables coordinates respectively. The flow patterns corresponding to the mth realization W m of Case 1 can be decomposed following Eq. (3.23). For the Case 2, it is required to use a set of basis functions

95 74 Flow-relevant ensembles in CLRM Figure 4.3: MDS plot. Color represent flow-based clusters. Numbers are assigned to all the realizations. Left: 5D tensor. Right: 6D tensor. for the variables coordinate to decompose the tensor S 2. In general for a 6D tensor, the flow patterns corresponding to the mth realization W m can be decomposed as: W m = I J Z K i=1 j=1 z=1 k=1 v=1 V αijzkvϕ m i ψ j ν z ω k υ v, (4.2) where K = K 1 and K = K 2 for the tensors S 1 and S 2 respectively, V = 2 is the number of variables (s o and p), {υ z } V v=1 are the set of basis functions for the variables coordinate, and αijzkv m := R r=1 σ ijzkrv χ r, e m. Similarly to (2.63), the flow patterns can be approximated by truncation of the sums in Eq. (3.23) and Eq. (4.2). Model clustering The low-dimensional tensor representation in (3.23) and the methods described in Section are used for model clustering. We have derived the set of tensors A 1 = {A 1 1, A 1 2,, A 1 R } and A2 = {A 2 1, A 2 2,, A 2 R } corresponding to the lowdimensional representations of the reservoir flow patterns in Case 1 and Case 2 respectively, for an ensemble of R = 103 realizations. Here, we use them for constructing clusters with similar reservoir flow patterns. The tensor k-means clustering algorithm described in Section has been used to classify the sets A 1 and A 2. The algorithm is provided with the number of clusters K c = 9, which have been selected by visual inspection of the MDS plots depicted in Fig To visualize the clusters, the MDS map is constructed using a distance matrix D based on the approximated dissimilarity function defined in (3.16). In the MDS plot, every dot corresponds to

96 4.2 Assessing the quality of the reduced-size ensembles 75 Figure 4.4: Snapshot of oil saturation front (top layer) of sample models from different clusters after 30 years of production. Background colors represent clusters Figure 4.5: Snapshot of oil saturation front (top layer) of sample models from different clusters after 30 years of production. Background colors represent clusters

97 76 Flow-relevant ensembles in CLRM a low-dimensional representation of the reservoir flow patterns for one realization, and the colors indicate clusters. Some of the characteristic reservoir flow patterns from the clusters in the right plot of Fig. 4.3 are depicted in Figs. 4.4 and Assessing the quality of the flow-relevant ensembles In this section, a reduced-size ensemble is created using the method described in Section 4.1. According to Yeh et al. [2014], one approach to determine the quality of a reduced ensemble generated by any discrimination method is to depict, for the reduced ensemble and a fixed input scenario, the CDF curves of a target variable (e.g., the total oil rates a particular time) and compare them with the CDF of the same target variable from the full ensemble. The criterion to determine the quality of the reduced ensemble says that, the closer in shape the CDF from the reduced ensemble is to the shape of the CDF from the full ensemble, the more representative the reduced ensemble is to approximate the dynamical features of the full ensemble. In this section, we use this criterion for assessing the quality of the reduced-ensembles. The random nature of the model selection method proposed in Section 4.1 implies that for every random seed, the method generates a new reduced ensemble, and therefore we obtain a new CDF of the total oil production. For instance, it is possible to visualize and characterize the spread of the CDFs that can be achieved with the method, if the process is repeated multiple times. In this context, a wider spread in the CDF plot indicates a poorer quality of the discrimination mechanism. In order to assess the quality of the selection method and the reduced ensembles based on flow attributes, we have reconstructed the CDFs of the total oil production at the end of the simulated production period t = 30 years, and we have used several flow indicators for model discrimination: the fingerprint technique using TOF information (following the work of Yeh et al. [2014]), the 5D reservoir flow patterns using only the oil saturation s o as target variable, and the 6D reservoir flow patterns using both the oil saturation s o and pressure p information. To exemplify, let us consider the cluster results presented in Section for the 5D and 6D tensors. In addition, we have used the fingerprints (TOF) to obtain clusters of models following the tensor workflow of flow characterization in Section and the tensor clustering algorithms presented in Section Once the models are clustered, the last step is the application of the method in Section 4.1 to generate a reduced ensemble and the CDF of the total oil rates associated to it. In the first experiment, we select the size of the flow-relevant ensemble to be R red = 15 realizations. The step that generates a reduced ensemble is repeated n = 1000 times in order to study the spread of the CDF of total oil production of the reduced ensemble, and to determine the quality of the flow-relevant ensembles. The results of the experiment are depicted in Fig The black line corresponds to the CDF of the total oil production using the full ensemble, which constitutes the target that we aim to approximate with the flow-relevant ensemble. The colored surfaces indicate the spread of the CDFs generated by the method in Section

98 4.2 Assessing the quality of the reduced-size ensembles 77 Figure 4.6: CDF of total oil production using 15 realizations. For every flow measure, the CFD is computed 1000 times. Surfaces show the dispersion around the real CFD (black line). Blue: Random sampling. Yellow: TOF. Red: 5D Tensor. Green: 6D Tensor. Black line: full ensemble. Figure 4.7: Variance of cum oil distribution for all the percentiles. 15 realizations

99 78 Flow-relevant ensembles in CLRM Figure 4.8: CDF of total oil production using 50 realizations. For every flow measure, the CFD is computed 1000 times. Surfaces show the dispersion around the real CFD (black line). Blue: Random sampling. Yellow: TOF. Red: 5D Tensor. Green: 6D Tensor. Black line: full ensemble. Figure 4.9: Variance of cum oil distribution for all the percentiles. 50 realizations

100 4.2 Assessing the quality of the reduced-size ensembles 79 Figure 4.10: CDF of total oil production using 85 realizations. For every flow measure, the CFD is computed 1000 times. Surfaces show the dispersion around the real CFD (black line). Blue: Random sampling. Yellow: TOF. Red: 5D Tensor. Green: 6D Tensor. Black line: full ensemble. Figure 4.11: Variance of cum oil distribution for all the percentiles. 85 realizations

101 80 Flow-relevant ensembles in CLRM 4.1. The blue surface corresponds to the spread of the CDFs due to generation of reduced ensembles using random selection of realizations from the full ensemble. It is clear that a random selection of the realizations without the clustering step creates a wider spread in the CDFs of total oil. For instance, a random selection of realizations without a proper clustering leads to a poor performance of the reduced ensemble in approximating the CDF of a target variable. Next we studied the spread of CDFs using flow indicators as discrimination method for clustering. The yellow surface covers the spread of the CDFs when using the fingerprint technique. Despite the spread being narrower than the one with random sampling, the method exhibits poor performance when approximating the CDF of the total oil production using the full ensemble (black line). The spreads resulting from approximating the CDF of the full ensemble using reservoir flow patterns are depicted in the red and green surfaces. From the results in Fig. 4.6, it is clear that the use of reservoir flow patterns has a big impact on improving the quality of the reduced ensembles, as they approximate the full CDF of total oil more accurately. Moreover, the results indicate that the more flow information we include in the description of the reservoir flow patterns, the more accurate the approximation of the CDF of total oil from the flow-relevant ensemble is. The quality of the reduced ensembles can be quantified by studying the variance of the spread of the CDFs in Fig For the case where R red = 15 realizations, the variance for all the percentiles associated with the different flow indicators is depicted in Fig The plots indicate that there is a large variance in approximating the CDF from the full ensemble for the initial range of percentiles, while the original CDF is better approximated by all the methods around the 50th percentile. The results show that using the reservoir flow patterns as dissimilarity measures reduces the variance of the spread of the approximated CDFs with approximately one order of magnitude. Moreover, it motivates the use of pressure information combined with oil saturation for the model discrimination. We have performed experiments for the case where R red = 50 and R red = 85 with similar results and conclusions, see Figures Acceleration of the parameter estimation stage using reduced-size ensembles Introduction In geostatistics, the uncertainty of the subsurface is accounted for by considering multiple realizations of the petrophysical parameters, and the parameter estimation loop of the CLRM updates all of these realizations based on field data such as well production data and time-lapse seismic information. Typically, a large number of geological realizations ( 10 3 ) are used, and the dynamical complexity of realistic reservoir models limits the possibility of performing parameter estimation (variational and ensemble methods) for multiple realizations. This fact motivates the de-

102 4.3 Acceleration of the parameter estimation stage using reduced-size ensembles 81 Parameter Value Units Simulation parameters Simulation time 30 [year] Time step 1 [month] Measurement errors, standard deviations Flow rates 12 [m 3 /day] Bottom-hole pressures 6 [bar] Fractional flow (watercuts) Parameter estimation experiment Observation time interval 3 [month] Update time interval 6 [year] Seismic data time interval 6 [year] End of history matching 18 [year] Table 4.2: Details of the experimental setting. sign of methodologies to accelerate the parameter estimation stage of the CLRM. In this section, we use the techniques developed in Chapter 3 and in Section 4.1 to perform parameter estimation of petrophysical properties (grid block permeabilities) and history matching (EnKF) of a realistic reservoir (the Brugge field) by means of reduced-size flow-relevant ensembles, which characterize the dominant reservoir flow patterns generated by flow simulation using the original ensemble Parameter estimation using flow-relevant ensembles Previously, we illustrated the benefits of implementing flow-based model selection for the construction of reduced ensembles that approximate more accurately the CDF of a target variable. In this section, we describe the application of flow-based model discrimination, presented in 4.1, for the parameter estimation of petrophysical properties using the EnKF. The Brugge field is produced for 30 years, with a time step of t = 1 month. The measurements to be considered are the total rates at the producer wells, the bottomhole pressure at the injector wells and the fractional flows at the producers, with an observation time interval of 3 months. In the experiment, measurement errors are modeled as white noise, i.e. stochastic variables with a Gaussian distribution of zero mean and properties defined in Table 4.2. In addition, seismic data is collected every 6 years, which provides information related to the front location. During the production stage, the petrophysical parameters are updated 3 times, with an interval of 6 years and the last update is made 18 years after production starts. The key parameters for the experiment are summarized in Table 4.2. For the parameter estimation, we have used the EnKF module for MRST (an implementation developed by TNO).We define realization number 104 to be the truth. In the experiment, we considered three types of ensembles:

103 82 Flow-relevant ensembles in CLRM Ensemble 1: The complete ensemble of R = 103 realizations. Ensemble 2: A reduced ensemble of R red = 15 realizations, constructed by random sampling of realizations from the full ensemble. Ensemble 3: A flow-relevant ensemble of R red = 15 realizations, constructed after the application of the workflow for flow characterization described in Section 3.4, with the extended reservoir flow patterns (6D tensor: s o and p), and the model discrimination presented in Section 4.1. The plots of production data (total flow rates) in several production wells, after the final update are presented in Figs All of the plots consist of three figures, each one associated with the flow rates in a particular well for the three ensembles. The history matching period, which is depicted with a green background, ends at 18 years. Thereafter, the reservoir models are used for prediction, during the period depicted with a pink background. Let us first analyze the results for Ensemble 1 (the full ensemble). The history matching results for most of wells shows a very good agreement of the model outputs with the data, as all the realizations resemble the output of the truth model, until the end of the history matching period. However, when models are used for prediction there is a huge mismatch between the predicted outputs of the ensemble and the truth. To exemplify, in Fig. 4.12, the water breakthrough at well 21 in the truth model happens around the day 7300, while the spread of predicted water breakthroughs of the ensemble ranges from 6400 days to 8800 days, with the majority of the realizations having a water breakthrough after 7900 days. In Fig. 4.13, the water breakthrough at well 23 in the truth model happens at 9900 days, and no realization of the updated ensemble is able to predict the phenomenon. In Fig. 4.14, the water breakthrough at well 24 in the truth model happens at 7050 days, while the predicted water breakthrough of all of the updated realizations ranges from 7400 to 8100 days. Fig shows that even when the water breakthrough information at day 3450 is captured during the period of history matching, there is a poor agreement between the predictions of the flow rates of the ensemble and the truth model. The results for Ensemble 2, created by a random selection of realizations from the full ensemble, shows similar results as the previous case, where there is a good history match, but the updated realizations have poor prediction capabilities. In Fig. 4.12, the predicted water breakthroughs at well 21 for the reduced ensemble happens around day 6550, which is 2 years earlier than the water breakthrough seen at well 21 in the truth model. In Figs and 4.14 the predicted water breakthroughs at well 23 and 24 of the reduced-ensemble also occur 600 and 400 days before the water breakthrough of the truth, respectively. For the Ensemble 3, there is an acceptable prediction of the flow rates. While the previous results are acceptable for history matching, they become attractive from the computational point of view. Both experiments were performed on a personal computer equipped with an 8 Core Intel i7-3.4ghz and 32GB of RAM. In this application, the parameter estimation experiment using the full ensemble

104 4.3 Acceleration of the parameter estimation stage using reduced-size ensembles 83 Figure 4.12: History matching and rate predictions. Total rates at well 21. Green background shows history matching interval. Pink background shows prediction interval. Figure 4.13: History matching and rate predictions. Total rates at well 23. Green background shows history matching interval. Pink background shows prediction interval.

105 84 Flow-relevant ensembles in CLRM Figure 4.14: History matching and rate predictions. Total rates at well 24. Green background shows history matching interval. Pink background shows prediction interval. Figure 4.15: History matching and rate predictions. Total rates at well 28. Green background shows history matching interval. Pink background shows prediction interval.

106 4.4 Production optimization with flow-relevant ensembles 85 took 5 days to be concluded, while the experiment using reduced-ensembles took 1day, reducing the computational effort with a factor of Discussion The computational effort for performing parameter estimation using the EnKF is proportional to the number of realizations, and the computational advantage of using a reduced-size ensemble is clear. The parameter estimation experiment presented in this section shows that the use of reduced-size flow-relevant ensembles reduces the computational cost of performing parameter estimation of realistically sized reservoir models using ensemble methods, while still keeping acceptable parameter estimation results, in terms of history matching and predictability of flow rates and water breakthroughs. One striking phenomena depicted in Figs , is the improved prediction capability of the flow-relevant reduced ensembles. These results can be interpreted from a probabilistic point of view, and here we elaborate on an explanatory hypothesis. Experience shows that despite that the realizations from the ensemble may look very different from a geological perspective, they may generate similar reservoir flow patterns. This observation leads to a simplification of the uncertainty description, because instead of dealing with thousands of geological realizations which describe the geological parameter (static) space, we create very few clusters of models with similar dynamical behavior. Provided that the full ensemble of realizations quantifies properly the petrophysical parameter space, there is a high probability for the truth model to be close to one realization (or many) from a dynamical point of view. This hypothesis needs to be further tested. After the clustering process, there are usually large clusters with more realizations than others. This suggest that there is a higher probability for the elements of the large cluster to have a similar dynamical behavior to the truth model. With the methodology proposed in Section 4.1, we choose a number of realizations that is proportional to the size of the cluster, and therefore it is likely to capture the dynamical behavior of the truth when selecting realizations from a large cluster. 4.4 Production optimization with flow-relevant ensembles In this section, the workflow presented in Section is used to cluster the geological realizations of a synthetic reservoir, in order to accelerate the computation of a robust production strategy that maximizes the financial performance through the life cycle of the reservoir.

107 86 Flow-relevant ensembles in CLRM Property Value Unit k md φ ρ w 1014 kg/m 3 ρ o 859 kg/m 3 µ w 1 cp µ o 5 cp S wc S or krw kro n w 4 - n o 4 - Table 4.3: Rock and fluid properties for the model in Section The reservoir model We consider an ensemble of 2D oil reservoirs with a square geometry of length L = 700m, one layer of 2m thick, produced with 4 wells (2 injectors, 2 producers). The numerical model of one realization has 441 grid blocks of size 21m 21m and the ensemble size R = 1000 realizations. A top view of some samples from ensemble, and well locations are shown in Fig The sequential solvers of MRST given by Lie et al. [2012], have been used to solve the pressure and saturation equations and the production has been simulated for a period of 1095 days, time step of 21.9 days. The adjoint module of MRST has been used to compute the optimal production strategies described in the next subsection. The physical parameters are presented in Table 4.3. Figure 4.16: Left: Permeability (md) and injectors/producers location for one realization. Right: Samples of the ensemble.

108 4.4 Production optimization with flow-relevant ensembles Generation of the flow-relevant ensembles The reduced-size flow-relevant ensemble is found by characterizing the saturation patterns of every reservoir simulation of the ensemble with the aid of tensor decompositions. This characterization generates compact representations of reservoir flow profiles which are used to measure the dynamical dissimilarity between reservoir models. With this approach we select, in the low-order space, realizations from a large ensemble that are dynamically/flow relevant in an optimized setting. In a dynamical environment (flow simulations), realizations have less variability than in a static setting (geological realizations). Hence, we construct a reduced-size ensemble of geological realizations that possesses a similar performance as the original ensemble for purpose of production optimization. Flow simulation The objective of this study is to classify the types of flow patterns in an ensemble of realizations in an optimized setting. The experiment consist of the simulation of the reservoir flow patterns of each of the realizations, K = 50 simulation steps, by using an optimal production strategy that maximizes the NPV in Eq. (2.37), with no discount factor. We have considered an ensemble of realizations with size R = 1000 and have used the model parameters presented in Table 4.3. Flow characterization and approximation The reservoir flow patterns are collected in a 4D tensor S of size I J K R where I = 21, J = 21, K = 50 and R = We use our in-house built SDM algorithm to decompose and characterize the 4D tensor formulation S of the reservoir flow patterns for every reservoir model of the ensemble. We perform modal truncation of the different dimensions with r = 10. Once reservoir flow patterns are represented in their tensor formulation, models are compared in terms of their dissimilarity of saturation patterns in the low dimensional space using the dissimilarity measure in Eq To visualize dissimilarity between the control-relevant saturation patterns from the whole ensemble of realizations in the low dimensional space, we use the MDS technique described in Section The MDS map is presented in Fig The realizations used for illustration of similar and dissimilar reservoir flow patterns in Figs and 4.18 are also indicated with numbers in the MDS plot in Fig In Fig we present the temporal evolution of saturation and NPV of three realizations of the ensemble with high dissimilarity in their saturation patterns. In Fig we present the temporal evolution of saturation and NPV of three realizations of the ensemble with low dissimilarity in their saturation patterns. The flow patterns and realization depicted in Fig 4.18 and 4.19 show that although realizations from the ensemble might look different from a geological perspective, their variation in terms of dynamical behavior in an optimized setting, i.e. saturation patterns, can be smaller.

109 88 Flow-relevant ensembles in CLRM Figure 4.17: MDS for the low dimensional flow profiles from the ensemble. Colors represent final NPV. Left: 3D map. Right: 2D map Representative models are sampled from the metric space map in order to reduce the ensemble size and to select the realizations that are significantly dynamically dissimilar. For this, the models are clustered using the methods described in Section 3.4.2, and R red = 50 representative models are selected using the method described in Section Production optimization under geological uncertainty using flow-relevant ensembles RO can be used to design production strategies for water flooding, when uncertainty is represented by an ensemble of geological realizations. In practice, models and ensembles might be large-scale, and RO techniques may not be computationally feasible or require a large amount of computational resources and time in order to obtain a production strategy. Yeh et al. [2014] have provided evidence that only a few flow-relevant realizations are needed for model optimization under uncertainty. In this Section, we demonstrate that the use of flow-relevant ensembles may increase the computational feasibility of techniques for production optimization under uncertainty. A production strategy that maximizes the average value of NPV using the controlrelevant ensemble is generated with RO techniques. We use a gradient-based methodology to compute the solutions, (see e.g. the work of Jansen [2011] for a complete review) with the objective function and gradients described in Eqs 2.37, 2.39 and (2.40). For the robust strategy we use the original (R = 1000) and the flow-relevant ensemble (R red = 50). The production strategies are presented in Fig

110 4.4 Production optimization with flow-relevant ensembles 89 Figure 4.18: Saturation patterns and NPV build-up for realizations with highly dissimilar flow. Figure 4.19: Saturation patterns and NPV build-up for realizations with similar flow.

111 90 Flow-relevant ensembles in CLRM Figure 4.20: Injection and production rates for RO. Blue/Red: Full ensemble (R = 1000). Black: Strategy with control-relevant ensemble (R red = 50) We simulate the full ensemble with the production strategies in Fig A distribution of the final NPV for the full ensemble with the two different strategies is presented in Fig After the clustering of realizations with similar flow-relevant reservoir flow patterns, the production strategy for the reduced ensemble in Fig resembles the production strategy generated using the full ensemble of realizations, and therefore both production strategies have a similar NPV distribution when applied to the full ensemble of realizations Discussion The application case discussed in the previous Section implies the computation of the optimal production strategy for every reservoir model of the ensemble. However, we may also consider the use of an unique production strategy to generate an ensemble of reservoir flow patterns, and select the geological realizations that prove to substantially contribute to the dynamical variability using the tensor-based methods described in the previous sections The Effect of the Production Strategy on the Generation of Flow-Relevant Ensembles In order to study and quantify the effects of different production strategies on the dynamical variability of the ensemble, we have generated reservoir flow patterns for

112 4.4 Production optimization with flow-relevant ensembles 91 Figure 4.21: Distribution of NPV over the ensembles. the ensemble by using an unique control input. We have performed Nominal Optimization over the ensemble with the two production strategies depicted in Fig. 4.22, which correspond to the optimal production strategies for the reservoir realizations 1 and 2. We have used the methodology described in the previous Section to characterize the ensemble of reservoir flow patterns in the low-dimensional space. This allows the construction of the MDS maps to visualize the dynamical variability of the ensemble. In Fig. 4.23, the MDS maps of the flow-relevant dissimilarity for the Nominal Optimization experiments are presented. We have indicated with a number the ensemble members that have been subject of study in the previous Section. The shapes of the MDS maps for both Nominal Optimizations and the MDS map in Fig present significant differences, however, we have found evidence of similar clustering patterns with respect to the dynamical behavior and financial performance in both experiments. To exemplify this observation, the models with similar dynamical behavior in Fig. 4.19, are again clustered in the MDS maps for the Nominal Optimization experiments in Fig. 4.23, indicating that the dynamical variability of the ensemble remains almost invariant for different choices of production strategies. For the MDS plots in Fig. 4.23, there is evidence of a connection between the reservoir flow patterns generated with the Nominal strategy and the NPV, as clusters of models with similar dynamical performance have also similar financial performance. This fact is a direct consequence of the uniqueness of the production strategy used for the simulation.

113 92 Flow-relevant ensembles in CLRM Figure 4.22: Left: Optimal production strategy for Reservoir model 1. Right: Optimal production strategy for Reservoir model 2. Blue: Injector rates 1. Green: Injector rates 2. Red: Producer rate 1. Cyan: Producer rate 2. Figure 4.23: MDS for Nominal Optimization. Left: Optimal production strategy for Reservoir model 1 applied to the ensemble. Right: Optimal production strategy for Reservoir model 2 applied to the ensemble. Color scale represent final NPV.

114 4.5 Concluding remarks Concluding remarks In this Chapter, we presented an application of the workflow for model clustering developed in Chapter 3 to reduce the computational complexity of the CLRM loops. Our approach was the selection of flow-relevant realizations to constitute a reducedsize ensemble, which is used for robust optimization and parameter estimation. First, we investigated the quality of flow-relevant ensembles, and we have demonstrated experimentally that the selection of realizations based on relevant flow attributes increases their quality with respect to the capability of reconstructing the CFD of a output variable. For the purpose of model-based recovery optimization under uncertainty, only a few flow-relevant realizations with distinctive flow properties are required to efficiently perform robust optimization. We show similar evidence for the parameter estimation step. The application of the methods described in Chapter 3 combined with the selection mechanisms of representative realizations have the potential to become a powerful tool to accelerate the optimization and parameter estimation loops under geological uncertainty in the context of practical CLRM.

115

116 Part II A tensor approach for the reduction of dynamical complexity 95

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118 5 CHAPTER Tensor methods for the model reduction of reservoir models 1 In this chapter, we propose a tensor framework for reduced order modeling in reservoir engineering, where tensor decompositions and representations of reservoir flow patterns are used to characterize empirical features of flow simulations. The concept of classical Galerkin projection is extended to perform projections of flow equations onto empirical tensor subspaces, generating in this way, reduced order approximations of the original mass and momentum conservation equations. The methodology is applied to compute gradient-based optimal production strategies for water flooding using tensor-based reduced order adjoints. 5.1 Introduction Simulation of multiphase flow through porous media is central for the modern practice of reservoir engineering. Currently, the increasing computational capabilities and the advent of smart field technologies allow the design of model-based operational strategies to maximize the financial performance during the life cycle of the reservoir. For this, numerical reservoir models with large number of states (in the order of ) are used as equality constraints in the optimization problem, see e.g. the works of Sarma et al. [2008a], Jansen et al. [2008] and Van den Hof et al. [2012] for future perspectives of the research field. Currently, the computational costs of reservoir simulations is too high to perform ensemble-based optimization and parameter estimation steps. Hence, there is a clear need for models with reduced dimensional complexity that can be used for efficient reservoir simulation. 1 Part of the content of this chapter is also published or presented in E. Insuasty, P. M. J. Van den Hof, S.Weiland, and J. D. Jansen. Tensor-based reduced order modeling in reservoir engineering: An application to production optimization. In 2nd IFAC Workshop on Automatic Control in Offshore Oil and Gas Production, Florianopolis, Brazil,

119 98 Tensor-based MOR In the last few decades, model reduction of linear systems has reached a high level of maturity, see e.g. Antoulas et al. [2001], and Proper Orthogonal Decomposition (POD) has been proved to be an efficient tool for model order reduction of large scale (non-linear) dynamical systems. In Chapter 2 we have introduced the concept of dynamical complexity of reservoir models and a review of the literature in the topic. Currently, the POD-based piecewise trajectory linearization (PTWL), see, e.g. Cardoso et al. [2009], is the leading technique for the reduction of the dynamical complexity of reservoir models. Its limitations were previously discussed in Chapter 1. To overcome the limitations of POD, we exploit the spatial-temporal nature of flow patterns, taking advantage of the spatial correlations that are usually lost during the computation of the classical POD projection spaces. In this chapter, we study the use of multidimensional arrays (tensors) as a natural representation of the flow solutions and empirical projections. For a complete overview of the available techniques for signal and system approximations using tensor decompositions, see Van Belzen and Weiland [2012]. Tensor algebra and analysis are largely unexplored topics in reservoir engineering. Afra and Gildin [2013, 2016], Afra et al. [2014] have introduced a reduced rank approximation of permeability fields using tensor decompositions. Insuasty et al. [2017b] have presented tensor characterizations of flow profiles and the evaluation of dissimilarity measures between reservoir models, with applications in the generation of flow-relevant ensembles, and Gildin and Afra [2014] and Insuasty et al. [2017a] have used low-dimensional tensor representations of parameter fields for parameter estimation. In this chapter, we introduce tensor-based model order reduction techniques in reservoir engineering, with applications in production optimization of water flooding. 5.2 Classical empirical projection techniques for model order reduction In this section we provide an introduction to system theoretical and experimental methods for model order reduction. Particularly, we focus on balanced truncations and POD, as classical methodologies for the reduction of dynamical complexity of reservoir models, defined as the dimension of the state vector of all pressures and phase saturations throughout the reservoir. For a detailed description of the philosophy and methods of classical Galerkin projections, we refer to the work of Kirby [2000]. A classical projection technique is the so called Balanced Truncation, see e.g. the work of Antoulas et al. [2001], which uses system properties such as controllability and observability, which are classically treated in a linear algebra setting Classical spectral expansions Let X R N be the spatial domain, and let ϕ i be the spatial mapping ϕ i : X R that describes a spatially distributed function. Let X be a separable Hilbert space of functions X R, with inner product, : X X R and norm ϕ i = [ ϕ i, ϕ i ] 1/2.

120 5.2 Classical empirical projection techniques for model order reduction 99 This makes (X, ) a normed space. For every ϕ j, ϕ k X we consider the inner product: ϕ j (ξ), ϕ k (ξ) = X ϕ j (ξ)ϕ k (ξ)dξ. (5.1) so that the induced norm becomes the standard L 2 norm of functions. That is [ 1 ϕ i 2 = X ϕ i(ξ) dξ] 2 2 and X = L 2 (X, R). In this context, the concept of orthonormality of different elements defining a subspace is defined. Let I Z be a (finite or infinite) index set of integers. A set of elements {ϕ i i I} with ϕ X is said to be orthonormal if: ϕ i, ϕ j = δ ij = { 1 if i = j 0 if i j (5.2) hold i, j I. In addition, the set of elements {ϕ i i I} is said to be an orthonormal basis of X if it constitutes a basis and if it is an orthonormal set. The linear span of all elements in such an orthonormal set is denoted span{ϕ i i I} and has a dimension equal to card(i) the cardinality of the index set I. Then every member f X admits an unique representation f = a i ϕ i (5.3) i=1 where a i = f, ϕ i and the equality (5.3) needs to be interpreted in the strong sense: lim I f I f, ϕi ϕi = 0. (5.4) i= POD basis functions For spatial-temporal systems, we consider signals that evolve over space X R N 1, as well as time T R. Let s : X T R be a continuous differentiable signal s(, t) X, where X = L 2 (X, R) for all t T, and let us assume the system is described by a set of partial differential equations R(s) = 0, (5.5) where R is a polynomial differential operator with differentiation in temporal and spatial variables which belongs to the same Hilbert space L 2 (X, R), and has the form R(s) = M(s) D(s), where the function M( ) is given by

121 100 Tensor-based MOR M = m 0 + m 1 t + + m nt n t, (5.6) nt t where m i R for i = 1,, n t and D is a differential operator that does not involve temporal derivatives. In reservoir engineering, the operator R( ) would represent oil saturation equation, and s would represent oil saturation. Let s(, t) X for all t < T T. If {ϕ i i I} is an orthonormal basis of X then any solution of (5.5) can be expanded as s(ξ, t) = i=1 a i(t)ϕ i (ξ), where a i (t) = s(, t), ϕ i is a time varying coefficient. Let s r (ξ, t) = r a i (t)ϕ i (ξ) (5.7) i=1 be the r th order approximation of s. Typically, R(s r ) 0, but the Fourier coefficients a i (t) in (5.7) can be selected to satisfy a system of ordinary differential equations in (a 1,, a r ), in such a way that the projection of R(s r ) onto the space spanned by {ϕ i i I} is equal to zero for all t 0. In this context, POD functions are of particular interest, mainly because they rely on experimental or simulation data from numerical models and have been successfully applied as part of model reduction techniques in fluid dynamics, structural vibrations, etc. Let s : X T R be given with s(, t) X, t < T T. The set of functions {ϕ i (ξ)} for i = 1, 2,, r is called a POD basis for an r-dimensional subspace of L 2 (X, R) associated with s, if it minimizes the following cost function: J(ϕ 1, ϕ 2,, ϕ r ) = T 0 s(, t) r 2 s(, t), ϕ i ϕ i dt i=1 s.t. ϕ i, ϕ j = { 1 if i = j 0 if i j 2. (5.8) The first POD basis function ϕ 1 maximizes the time averaged projection of the elements of s(, t) onto ϕ 1. This optimization problem can be formulated as follows: ϕ 1 = arg max = ϕ 1 L 2(X,R) T 0 s(, t), ϕ 1 2 dt ϕ 1 2, (5.9) 2 which can be solved using calculus of variations. If the set of functions {ϕ i (ξ)} is an orthonormal set, the expression in (5.9) can be calculated by finding an extremum of the following augmented cost function as follows: J 1 (ϕ 1 ) = T 0 s(, t), ϕ 1 2 dt λ 1 ( ϕ ), (5.10)

122 5.2 Classical empirical projection techniques for model order reduction 101 where λ 1 is a Lagrange multiplier. The first necessary condition for an extremum of (5.10) is that d dδ J 1(ϕ 1 + δψ) = 0, where δ R and ψ L 2 (X, R). By expanding the expression in Eq. (5.10) we get: d dδ J 1(ϕ 1 + δψ) = d [ T dδ 0 = d [ T dδ = d dδ 0 [ T 0 ] s(, t), ϕ 1 + δψ s(, t), ϕ 1 + δψ dt d [λ 1 ϕ1 + δψ, ϕ 1 + δψ ] dδ )( ) ] ( s(, t), ϕ 1 + s(, t), δψ s(, t), ϕ 1 + s(, t), δψ ( )] [λ 1 ϕ 1, ϕ 1 + ϕ 1, δψ + δψ, ϕ 1 + δψ, δψ d dδ ( s(, t), ϕ s(, t), ϕ 1 s(, t), δψ + s(, t), δψ 2) ] dt d dδ ( )] [λ 1 ϕ 1, ϕ 1 + ϕ 1, δψ + δψ, ϕ 1 + δψ, δψ, dt for instance, the first condition of optimality reads: for instance, T d dδ J 1(ϕ 1 + δψ) = 2 s(, t), ϕ 1 s(, t), δψ dt 2λ 1 ϕ 1, δψ 0 T = 2 s(, t), ϕ 1 s(, t)dt λϕ 1, ψ = 0, (5.11) 0 T 0 s(, t), ϕ 1 s(, t)dt λϕ 1 = 0 (5.12) is the necessary condition for an extremum of Eq. (5.10). Let us define the correlation operator Φ : L 2 (X, R) L 2 (X, R) as Φ(ψ) = T s(, t), ψ s(, t)dt. The 0 correlation operator Φ is a well defined linear, bounded and non-negative operator on L 2 (X, R). If card(x ) <, the correlation operator Φ becomes a symmetric, nonnegative matrix and its elements represent the correlation between the collection of signal samples in a discretized spatial-temporal domain. The problem of finding the first POD basis function is equivalent to solving the singular value (or eigenvalue) problem for the correlation operator Φ: Φ(ϕ 1 ) = λ 1 ϕ 1. (5.13) The second POD basis function ϕ 2 can be calculated by including the orthogonality constraint to the first POD basis function ϕ 1, and so on for the next POD basis

123 102 Tensor-based MOR functions. Then, the POD basis correspond to the eigenfunctions of the correlation operator Φ. Suppose that rank(φ) r, then the set {ϕ 1, ϕ 2,, ϕ r } is a POD basis if and only if Φϕ i = λ i ϕ i for i = 1, 2,, r and λ 1 λ 2 λ r The Galerkin projection Let us define the finite dimensional functional space X r := span{ϕ i i I} for the index set I := {1, 2,, r}. Let us define s r in (5.7) as the approximate solution of order r for the partial differential equation in (5.5), if, for all time t 0 the operator R( ) over s r (ξ, t) is orthogonal to the subspace X r, i.e., R(s r ) X r. The Galerkin projection reads: ( r ) ϕ i, R(s r ) = ϕ i, R a j (t)ϕ j (ξ) = 0 for i I (5.14) j=1 The Galerkin projection states that the approximated differential operator R(s r ) is orthogonal to the r dimensional subspace X r spanned by the subset of basis functions {ϕ i } for i = 1, 2,..., r so the approximation of R(s r ) in the span of the first r basis is exactly zero, see e.g. the work of Astrid [2004]. The expression in (5.14) defines a reduced order approximation of (5.5). 5.3 Tensor-based reduced order modeling The previous section described classical methods for model reduction of nonlinear systems. In this section, we introduce an algorithmic formulation for the classical Galerkin projection of dynamical systems onto empirical tensor spaces. We extend the concept of orthogonal projection of dynamical systems to the case where the empirical projection spaces span every coordinate of the spatial domain. The techniques described in this section are used for the reduction of the dynamical complexity of numerical hydrocarbon reservoir models Tensor representations and decompositions Let us consider the discretized spatial domain with Cartesian structure X = X 1 X N 1, such that card(x i ) = R i i.e., X i = {ξ (1) i,, ξ (Ri) i } for i = 1,, N 1 a finite set of spatial grid points, and the temporal domain T = {t (1),, t (R N ) }. Let us assume the sampled solution of (5.5), i.e. the oil saturation, is a mapping s : X T R. Let the vector spaces X i := R Ri for i = 1,, N 1 and T = X N := R R N be equipped with the standard Euclidean inner product. Without losing generality, let us assume that the signal s has only 1 component, i.e., only 1 dependent variable. Let x i X i for i = 1, N, then s has an associated multilinear mapping S : X 1 X N 1 T R defined with respect to the canonical basis as:

124 5.3 Tensor-based reduced order modeling 103 S(x 1,..., x N ) = R 1 l 1=1 R N l N =1 and has a representation of the form: s l1 l N e 1 (l 1), x 1 e 2 (l 2), x 2 e N (l N ), x N, (5.15) S = R 1 l 1=1 R N l N =1 s l1 l N e 1 (l 1) e N (l N ) (5.16) where the coefficient s l1 l N = s(ξ (l1) 1,, ξ (l N 1) N 1, t (l N ) ) takes the value of s at the spatial grid point ξ = (ξ (l1) 1,, ξ (l N 1) N 1 ) and time t = t (l N ). The denotes the tensor or outer product over a vector space. Suppose that for i = 1,, N, every (X i,, i ) is spanned using a different set of orthonormal basis functions {ϕ (lj) i } Ri j=1, i.e., ϕ (lj) i, ϕ (l k) i i = δ jk, where δ jk is the Kronecker delta function. Then, with this change of basis, the tensor S admits a representation of the form: S = = R 1 l 1=1 R 1 l 1=1 R N l N =1 R N l N =1 σ l1 l N ϕ (l1) 1 ϕ (l N ) N σ l1 l N Φ (l1l2 l N ) (5.17) where Φ (l1l2 l N ) = ϕ (l1) 1 ϕ (l N ) N is a rank 1 tensor, and σ l 1 l N is the (l 1 l N ) element of a core tensor σ l1 l N = S(ϕ (l1) 1,, ϕ (l N ) N ) = R 1 m 1=1 R N m N =1 s m1 m N e 1 (m 1), ϕ (l1) 1 e N (m N ), ϕ (l N ) N (5.18) Nested Galerkin projections Reservoir simulators describe the evolution of saturation and pressure over time and space X, which is typically represented with three dimensions (N = 3) in the Cartesian plane. For this reason, tensors are a natural way to represent and analyze data from spatial-temporal models. In this subsection, we can extend the concept of Galerkin projection onto the whole spatial domain, to a projection onto sets of orthonormal basis that span empirical spaces for the different dimensions of the spatial domain.

125 104 Tensor-based MOR Let us represent the solution of (5.5) in terms of s(, t) at the spacial-temporal index location (k 1, k 2,, k N ) using tensor expansions as follows: s k1 k N = = R 1 l 1=1 R 1 l 1=1 R N l N =1 R N 1 l N 1 =1 σ l1 l N ϕ (l1) 1, e 1 (k 1) ϕ (l N ) N, e N (k N ) α l1 l N 1 ϕ (l1) 1, e 1 (k 1) ϕ (l N 1) N 1, e N 1 (k N 1 ) (5.19) where the coefficients α l1 l N 1 in (5.19) are α l1 l N 1 = R N l N =1 σ l1 l N ϕ (l N ) N, e N (k N ). (5.20) Given sets of orthonormal basis functions for every spatial dimension {ϕ (li) i } i=1 Ri, we enforce the residual equation R(s) to be orthogonal to the basis functions that span every coordinate, then, a nested Galerkin projection of (5.5) over the Tensor projection spaces can be defined as ϕ l1 1, ϕ l2 2,, ϕ (l N 1) N 1, R(s) = 0 (5.21) N where, i is the inner product defined for the i-th dimension of the spatial coordinate, see Van Belzen and Weiland [2012]. The expression in (5.21) defines an ordinary differential equation that represents the projection of R(s) onto the basis 2,, ϕ (l N 1). The reduced order approximation of the dynamical system ϕ (l1) 1, ϕ (l2) N 1 in (5.5), can be found by generating ODEs from (5.21) by letting the basis indexes varying as l 1 = 1,, r 1, l 2 = 1,, r 2 until l N 1 = 1,, r N 1, where r 1 R 1, r 2 R 2 until r N 1 R N 1. In summary, the assumption of a Cartesian spatial domain X and temporal domain T associates a multidimensional array S to the solution of a dynamical system s. Then, a new representation of S in terms of rank-1 tensors allows the construction of empirical basis for every coordinate, which leads to the definition of a tensor expansion of s. Then, we use sequential projections onto the empirical basis that span every X i in order to generate a reduced order model. Moreover, it is theoretically possible to perform approximations in the time coordinate too, however, the physical interpretation and significance need to be investigated.

126 5.3 Tensor-based reduced order modeling Tensor-based model order reduction of a single-phase reservoir model In order to exemplify the method, let us consider the case of a model for the 2 dimensional single phase flow through porous media. Let the differential operator in Eq. (5.5) be: R = [ φ o c t t + κ x µ 2 x 2 + κ y µ 2 ] y 2, (5.22) then, the pressure dynamics over the spatial domain is described by the diffusion equation R(p) = 0: p R(p) = φ o c t t + κ x 2 p µ x 2 + κ y 2 p = 0, (5.23) µ y2 where the spatial domain has a Cartesian structure X = X 1 X 2, such that card(x 1 ) = I and card(x 2 ) = J i.e., X 1 = {ξ (1) 1,, ξ(i) 1 } and X 2 = {ξ (1) 2,, ξ(j) 2 }. Moreover, X = X 1 X 2 is the result of a Cartesian product of vector spaces (X 1,, 1 ) and (X 2,, 2 ), where X 1 := R I and X 2 := R J. Now, let us assume that the spatialtemporal solution of Eq. 5.23, i.e., p(x, y, t) has a spectral expansion with discretized spatial domain: p = I J l 1=1 l 2=1 a l1l 2 (t)ϕ (l1) x ϕ (l2) y (5.24) The we perform a Galerkin projection of Eq into the spaces spanned by the set of orthonormal basis {ϕ (i) x } and {ϕ (j) y } for i = 1,, I and j = 1,, J, using the nested structure of inner products. Therefore a differential equation for the Fourier coefficient a ij (t) is given by: φ o c t ȧ ij (t) = κ x µ I l 1=1 a l1j(t) ϕ (l1) x, ϕ (i) x + κ y µ J l 2=1 a il2 (t) ϕ (l2) y, ϕ (j) y (5.25) The mathematical derivations are developed in the following section Application of the nested Galerkin projection We define a dynamical system in Eq. (5.23) using Galerkin projections onto every basis of the different spatial coordinates of the problem in consideration.

127 106 Tensor-based MOR Projection onto the x coordinate First, let us define the modal projection operator onto the x coordinate as: f(x)g(y), h(x) x = f(x), h(x) g(y). (5.26) Now, let us perform the modal projection of Eq. (5.23) onto the basis functions {ϕ x (i) }. For every pair of basis (ϕ (i) x, ϕ (j) y ) we obtain an expression of the form: R(p), ϕ (i) x = p φ x o c t t + κ x 2 p µ x 2 + κ y 2 p + q, ϕ(i) µ y2 x. (5.27) x By using the spectral expansion defined in Eq. (5.24), we obtain: R(p), ϕ (i) x x = R ( I J l 1=1 l 2=1 a l1l 2 (t)ϕ (l1) x ϕ (l2) y ), ϕ (i) x x, (5.28) which is a residual expression in (y, t) of the form. Let us perform the algebraic operations separately of each term in Eq. 5.28: p φ o c t t, ϕ(i) x = x φ o c t I J l 1=1 l 2=1 ȧ l1l 2 (t)ϕ (l1) x ϕ y (l2), ϕ (i) x x (5.29) Because of orthonormality of the basis functions, we have that p φ o c t t, ϕ(i) x = J φ o c t l 2=1 ȧil 2 (t)ϕ (l2) y, for l 1 = i 0, for l 1 i (5.30) The second term is expanded as κ x 2 p µ x 2, κ x ϕ(i) x = µ x The third term is expanded as = κ x µ I J l 1=1 l 2=1 I l 1=1 l 2=1 a l1l 2 (t) ϕ (l1) x J a l1l 2 (t) ϕ x (l1), ϕ (i) x ϕ y (l2), ϕ (i) x x x ϕ (l2) y (5.31)

128 5.3 Tensor-based reduced order modeling 107 κ y 2 p µ y 2, κ y ϕ(i) x = µ x I J l 1=1 l 2=1 a l1l 2 (t)ϕ (l1) x ϕ y (l2), ϕ (i) x κ y J µ l a 2=1 il 2 (t) ϕ y (l2), for l 1 = i =. (5.32) 0, for l 1 i x For instance, the modal projection of Eq. (5.23) onto the basis functions {ϕ (i) x } leads to R(p), ϕ (i) x = x J l φ 2=1 oc t ȧ il2 (t)ϕ (l2) y κ x I J µ l 1=1 l a 2=1 l 1l 2 (t) + κy µ a il 2 (t) ϕ (l2) y + κx I J µ l 1=1 l a 2=1 l 1l 2 (t) ϕ (l1) x, ϕ (i) x ϕ (l2) y, for l 1 i x ϕ (l1) x, ϕ (i) x ϕ y (l2), x for l 1 = i (5.33) Projection onto the y coordinate Now, the expressions in Eq (5.33) are projected onto the basis functions {ϕ (j) y }. Let us consider the case where l 1 = i in Eq (5.33), and let us project the first sum term: J φ o c t ȧ il2 (t)ϕ (l2) y + κ y µ a il 2 (t) ϕ y (l2) l 2=1 φ o c t ȧ ij (t) + κy J µ l a 2=1 il 2 (t) ϕ y (l2) = κ y J µ l a 2=1 il 2 (t) ϕ (l2) y, ϕ (j) y, ϕ (j) y, ϕ (j) y y for l 2 = j. (5.34) for l 2 j Now, let us consider the projection of the second sum term in Eq (5.33) onto {ϕ (j) y }: κ x µ I J l 1=1 l 2=1 a l1l 2 (t) ϕ (l1) x, ϕ (i) x ϕ (l2) y, ϕ (j) y y

129 108 Tensor-based MOR = κ x I µ l a 1=1 l 1j(t) ϕ (l1) x, ϕ (i) x for l 2 = j 0 for l 2 j. (5.35) For the case where l 1 i in Eq (5.33), the results are equal to the expressions in Eq. (5.35). In order to derive a differential equation for the Fourier coefficient a ij (t), we consider the case where l 1 = i and l 2 = j and the Nested Galerkin projections described in Eq. (5.21). For instance R(p), ϕ (i) x x, ϕ(j) y y = φ o c t ȧ ij (t) + κ y µ J l 2=1 a il2 (t) ϕ (l2) y, ϕ (j) y + κ x µ I l 1=1 a l1j(t) ϕ (l1) x, ϕ (i) x = 0, (5.36) for i = 1,, I and j = 1,, J. 5.4 Application case: Tensor-based model order reduction for multiphase flow models In this section we present an application case for the techniques illustrated in Sections 5.2 and 5.3. Classical POD and Tensor MOR techniques are applied for the reduction of dynamical complexity of a reservoir model The model We use a numerical model for oil-water fluid flow through heterogeneous porous media. The implicit solvers of MRST, see Lie et al. [2012], have been used to solve the pressure and saturation equations: v = λ t K p, v = q φ t s + (f w(s)(v + g d)) = q w ρ w (5.37) where v is the Darcy velocity, K the permeability tensor, p the pressure, q volumetric rates, φ the porosity, s the water saturation, q w water volumetric rates, ρ w the water density, g the gravity, d depth below the surface and f w (s) the water fractional flow

130 5.4 Application case: Tensor-based model order reduction for multiphase flow models 109 defined in terms of the relative permeabilities k rw and k ro as: f w (s) = k rw (s)/µ w k rw (s)/µ w + k ro (s)/µ o (5.38) and µ w and µ o the water and oil viscosities. We assume zero flow at the boundaries. We consider a reservoir with square geometry of side length L = 600m, one layer of 4m thick, 5 wells (1 injector, 4 producers). A top view of well locations, permeability and porosity fields are provided in Fig The numerical model has 900 grid blocks of size 20m 20 4m and the physical parameters are presented in Table 5.1. Figure 5.1: Permeability, porosity for the test case. The controls for this reservoir are the injection rate and the bottom hole pressures at producers Snapshots generation and decomposition A production strategy (Initial Schedule, Fig. 5.4) is used to generate K = 48 snapshots of pressure and saturation profiles (I = J = 30) and the variables are stored in separate multi-linear arrays: S = l 1=1 l 2=1 l 3=1 s l1l 2l 3 e 1 (l 1) e (l2) 2 e (l3) 3, P = l 1=1 l 2=1 l 3=1 p l1l 2l 3 e 1 (l 1) e (l2) 2 e (l3) 3, (5.39) where the coefficients s l1l 2l 3 and p l1l 2l 3 take the value of s and p at the spatialtemporal grid point of index (l 1, l 2, l 3 ).

131 110 Tensor-based MOR Next, the POD and tensor basis, i.e., oil saturation {ϕ (i) 1s {ϕ (i) 1p } and {ϕ(j) 2s }, and pressure } for i, j = 1,, 30, are computed by using decomposition algo- } and {ϕ(j) 2p rithms described in Subsections (SVD) and (SDM). Let us define the ij th tensor spatial pattern for saturation as ϕ j 1s ϕj 2s, which can be interpreted as an spatial pattern which contains some relevant directions in the snapshot data. To illustrate, some POD and tensor spatial patterns for saturation are presented in Fig Figure 5.2: POD and Tensor basis for saturation Model reduction The number of coordinates for this application case is N = 3, two corresponding to the spatial domain X = X 1 X 2, and one coordinate for time. POD models are obtained by projecting the (5.37) onto the first n s = 47 and n p = 20 POD basis for saturation and pressure respectively. For the Tensor-based case, we compute the spatial patterns ϕ j 1s ϕj 2s and ϕj 1p ϕj 2p for i, j = 1,, 7, which leads to a total of 49 spatial patterns for each dependent variable s and p. Similar to the POD case, let us select the first n s = 47 and n p = 20 spatial patterns for saturation and pressure respectively for the projection of (5.37). Production was simulated for a period of 3100 days, with a time step of 64.5 days. Some temporal snapshots of oil saturation profiles for the full order and reduced models are presented in Fig 5.3. On one side, it is clear that the POD approximation is not capable of generating trajectories with

132 5.5 Tensor-based reduced-order adjoint models for production optimization 111 physical significance, and the diffusive-convective nature of the full equations is lost after the projection onto POD subspaces. Figure 5.3: Oil saturation time snapshots for full model and reduced order model. In this application case, only 5 10% of computational gain was obtained by employing both POD and Tensor methods for MOR. This is a known limitation of every projection-based method, and it is due to the operations associated with the inner products in (5.21) at every iteration of the nonlinear solver. See Cardoso et al. [2010a] for methods to speed-up the procedure. 5.5 Tensor-based reduced-order adjoint models for production optimization Gradient-based production optimization of water flooding is a computationally expensive process that requires several forward reservoir and backward adjoint simulations, and there is a potential use of reduced-order modeling to accelerate this step

133 112 Tensor-based MOR Property Value Unit k md φ ρ w 1014 Kg/m 3 ρ o 859 Kg/m 3 µ w 1 cp µ o 5 cp S wc, S or krw kro n w, n o 3 - Table 5.1: Physical parameters of CLRM scheme. In this section, we compute production strategies using a modelbased approach. The computed production strategies are the ones that maximize financial performance (NPV) over the life cycle of the reservoir, and the models used to compute those production strategies are the full order model, a POD reduced order model, see e.g.,van Doren et al. [2006], and the reduced-order model based on projections onto spatial patterns from tensor decompositions. We study the stability of models, and compare the economic performance of the reservoir under the computed production strategies Production optimization In this work, we compute the optimal production strategy that maximizes NPV as it is described in Section using gradient-based methods that use the adjoint equations, see the Section and we refer to the work of Sarma et al. [2008a] and Jansen [2011] for a detailed description of the theory and algorithms Reduced order adjoint equations The adjoint-based approach for production optimization is one of the most efficient methods, as it only requires one simulation for the forward and adjoint model. The computation of the corresponding Lagrange multipliers still depends on the computation of the sensitivities of the full order model with respect to its states. As it is presented in Van Doren et al. [2006], a reduced order version of the adjoint model is obtained by projecting the adjoint model equations onto empirical low order subspaces, and with the aid of reliable reduced-order models we can reformulate an approximation of the adjoint equations. Let x be the states vector and let us consider the reservoir simulator in its residual formulation R(x) = 0, see, e.g., Eq. (5.5), or as it is described in Eq. (2.21). Then, the

134 5.5 Tensor-based reduced-order adjoint models for production optimization 113 Figure 5.4: Optimal production strategies for full order and reduced order approximations. adjoint model has the form: λ(k) R(k 1) x(k) = λ(k + 1) R(k) x(k) J(k) x(k), (5.40) where J(k) is the objective function to be maximized. In order to compute production strategies that maximizes a financial measure using the adjoint method, the adjoint model of the reservoir runs backwards in time to compute the required Lagrange multipliers of the optimization problem. To compute a reduced-order adjoint model, let us define the linear state transformation x = Ψa, then the reservoir equations on the transformed states are R(Ψa) = 0. Let Ψ R N Nr, with N r N, the projection of Eq. (5.40) onto the column space of Ψ is ˆλ(k) R(k 1) Ψ Ψ = ˆλ(k + 1) Ψ R(k) x(k) x(k) Ψ J(k) Ψ. (5.41) x(k) The reduced-order adjoint in Eq. (5.41) has been used to accelerate the production optimization step, see, e.g., Van Doren et al. [2006] for details. In this application case, we have used the automatic differentiation capabilities of MRST, see, e.g., Lie et al. [2012], to implement Eq. (5.41) which is required to maximize the NPV defined in Eq. (2.37) using the adjoint method described in Section

135 114 Tensor-based MOR Figure 5.5: NPV performance of the virtual asset with the different strategies Results We compute optimal production strategies for the POD and Tensor reduced order models with the methods described in Sections and 5.5.2, where reduced-order models and adjoints are used for forward and backward simulation. The results are illustrated in Fig In order to perform the projections in (5.41), Ψ and ˆR were computed for both POD and the tensor approach and coupled with the adjoint formulation provided by the fully implicit solvers of MRST. The production strategies consist of 5 control steps, and the corresponding NPV build-ups for the full model operated with those strategies are presented in Fig On one hand, it is illustrated that despite POD reduced-order models can be used for the purpose of optimization, their limitations in terms of approximation accuracy constrain the economic performance of the resulting optimal production strategy. On the other hand, the accuracy level achieved with tensor-based reduced order models affects positively their use in production optimization. In this application case, we show empirical evidence that the tensor strategy displays a better economic performance compared to the POD strategy, and that the tensor one is close to the optimal strategy for the full order model.

136 5.6 Concluding remarks Concluding remarks In this Chapter, tensor representations of flow profiles are used to generate empirical spaces where the equations of two-phase flow through porous media are projected independently in every physical dimension using the concept of nested Galerkin projections. For the application case presented in this chapter, tensor models are able to approximate accurately most of the dynamical characteristics of the full order model, while the POD case experiences the limitations reported in literature. However, the accuracy of tensor models is subject to the scope of the experiment design for the generation of empirical projection spaces. The computational gains are low compared to the full order model, which is a well reported limitation of the projection methods for model order reduction, and there exist methods in literature to accelerate nonlinear reduced order models, see, e.g., Cardoso et al. [2010a]. The advantages of using tensor representations for reduced order models lie in the higher levels of approximation accuracy that can be achieved compared with the classical POD models. As tensor methods are used to obtain th projection spaces, the tensor-based most relevant flow patterns are different from the ones obtained with the classical POD method, and we have displayed evidence that these methodologies can be used to generate reduced order models with better approximation quality, which can be potentially used in optimization routines. However, the preservation of stability is not granted by the method itself, and this is a theoretical aspect to be investigated in the future.

137 116 Tensor-based MOR

138 Part III A tensor approach for the reduction of geological complexity 117

139

140 6 CHAPTER Estimation of petrophysical parameters using tensor representations 1 Computer-assisted history matching (CAHM) has been successfully applied to reduce the level of uncertainty of the petrophysical properties by incorporating datadriven methods for parameter estimation. In this context, CAHM aims to assimilate production data by finding an updated set of model parameters (petrophysical parameters like permeabilities, porosities, net-to-gross ratios, etc.), such that the mismatch between production data and the model response is minimized. This has been a very prolific field of research and many algorithms have been developed in the last decades. For overviews we refer to the textbook by Oliver et al. [2008] and the review paper by Oliver and Chen [2011]. Here we will use the Ensemble Kalman filtering method for which we refer to the textbook of Evensen [2009] and the review paper by Aanonsen et al. [2009]. Typically, the number of parameters to be estimated exceeds by far the number of available field measurements, and the parameter estimation problem becomes illconditioned, see e.g. Oliver et al. [2008]. As the parameter estimation procedure is posed as an ill-conditioned optimization problem, there are usually several optimal solutions, i.e., there are multiple realizations of the petrophysical parameters that minimize the mismatch between the model output and the data. This implies the fact that while two history-matched reservoir realizations may look very different from the geological perspective, they may have very similar outputs in their response space. This phenomenon is illustrated in Fig From a Systems and Control perspective, a mathematical model that, when provided with different parameter sets, generates similar responses is usually classified 1 Substantial content of this chapter is also published or presented in E. Insuasty, P. Van den Hof, S. Weiland, J. Jansen, et al. Low-dimensional tensor representations for the estimation of petrophysical reservoir parameters. In SPE Reservoir Simulation Conference. Society of Petroleum Engineers,

141 120 Estimation of petrophysical parameters using tensor representations Figure 6.1: Rock permeability and well configuration of two updated reservoir models with very similar dynamical response (solid and dashed lines), adapted from van Essen et al. [2016]. as a non-identifiable model. Typically, reservoir models are non-identifiable, see e.g. Van Doren et al. [2009, 2011]. In simple words, the petrophysical parameters are currently described at a level of detail which is too high to achieve a proper validation from the data. This problem is usually addressed through the use of a prior term with spatial correlations described with two-point statistics (a covariance matrix). However, spatial features that extend over many grid blocks, such as channels or faults, are not captured by such a description. In order to address this issue, several authors have developed parameterization methods of geological structures which include: Zonation: Jacquard et al. [1965], Jahns et al. [1966] Pilot points: Bissell et al. [1997], De Marsily [1984]. Principal Component Analysis: Gavalas et al. [1976], Oliver [1996], Reynolds et al. [1996], Tavakoli et al. [2010]. Wavelets: Awotunde and Horne [2013], Lu et al. [2000], Sahni et al. [2005]. Level sets: Dorn and Villegas [2008]. Channel parameterizations: Van Doren et al. [2011]. Discrete Cosine Transform: Jafarpour et al. [2009, 2010]. Pluri-Principal Component Analysis: Chen et al. [2016]. Optimization-based PCA: Vo and Durlofsky [2014]. Recently, there have been several developments aimed at preserving geological realism after the CAHM step, see, e.g., Vo and Durlofsky [2015]. Sarma et al. [2008a] have introduced the kernel PCA method, which can capture, to some extent, higherorder statistics, but involves the solution of a nontrival pre-imaging problem, see also e.g. Ma and Zabaras [2011]. From the perspective of CAHM, parameterizations are attractive because:

142 6.1 Tensor modeling and decompositions for the representation of petrophysical properties Identifiability: They increase the identifiability of the inverse modeling problem by estimating fewer parameters. 2. Spatial structure: They sometimes maintain spatial features such as channels or faults, that are not captured by two-point statistics (covariances). Despite of these efforts, finding natural and simple representations of geological structures is still an active field of research. Tensor methodologies for parameter estimation constitute a relatively new subject in the reservoir simulation community, and Afra and Gildin [2013, 2016], Gildin and Afra [2014] and Afra et al. [2014] have explored the application of tensor decompositions to represent permeability fields for parameter estimation. In this chapter, we build on their work to achieve better predictive capabilities of updated channelized reservoir models in CAHM, while providing an explicit formulation of the tensor methodology for the parameterization of petrophysical parameters. We pursue this through the adaptation of an algorithm for parameter estimation that uses the low-dimensional tensor formulation of channelized permeability fields. In the next sections, we introduce the fundamental tensor tools, the type of parameterizations, the adapted ensemble algorithm for parameter estimation and a discussion of the results of the application of these developments in a test case. 6.1 Tensor modeling and decompositions for the representation of petrophysical properties Let us consider a reservoir with a 2D rectangular geometry with I J grid cells. Traditionally, the representation of the petrophysical parameters (permeability, porosity, etc.) has been in the form of an (I J 1) vector. Alternatively, if the spatial grid has a Cartesian structure, and if there is an ensemble of models, one can collect R realizations θ of size I J, and represent this data object in a three-dimensional array S of size I J R. That is, the ensemble of realizations is represented as a multidimensional array S R I J R. Such a multidimensional array is called a tensor and can be viewed as the natural generalization of vectors and matrices to higher dimensional objects. For an ensemble of 2D channelized permeability fields, a three-dimensional array is schematically depicted in Fig In the tensor representation, the geometrical and spatial structures of the petrophysical parameters remain intact. In addition, the multilinear domain allows extensions of classical matrix decompositions to tensors, and the notions of approximations and low-rank approximations can be also defined in the tensor framework. Formally, let us consider the spatial domain with Cartesian structure X = X 1 X 2, such that dim(x 1 ) = I, dim(x 2 ) = J, i.e., X 1 = {ξ (1) 1,, ξ(i) 1 } and X 2 = {ξ (1) 2,, ξ(j) 2 } a finite set of spatial grid points. Let the coordinate for the realizations be W = {w (1),, w (R) } and dim(w) = R. Following the definitions provided in Section 2.4, the tensor S constitutes a multilinear mapping S : X 1 X 2 W R, which can be represented with respect to canonical bases as:

143 122 Estimation of petrophysical parameters using tensor representations Figure 6.2: Schematic of the tensor representation of a petrophysical property (permeability) of an ensemble of channelized models. The horizontal axes represent spatial coordinates and the vertical axis the model coordinate. S = I J i=1 j=1 r=1 R θ ijr e (i) 1 e (j) 2 e (r) 3, (6.1) where the coefficient θ ijr = s(ξ (i) 1, ξ(j) 2, w(r) ) takes the value of θ w, the petrophysical property in consideration, at the spatial grid point ξ = (ξ (i) 1, ξ(lj) 2 ) of the reservoir realization of index w = w (r). The vectors e 1 (w), e 2 (w), e 3 (w) are the w th standard unit vectors in R I, R J, R R respectively. By using a different set of basis functions to span the functional spaces for each coordinate (see Section 2.4 for details) the tensor S admits a representation of the form S = I i=1 J j=1 R r=1 σ ijrϕ i ψ j ϑ r, where {ϕ i } I i=1, {ψ j} J j=1, {ϑ r} R r=1 are sets of orthonormal basis for the vector spaces X 1, X 2 and W respectively Tensor approximations of permeability fields In order to illustrate the use of tensor decompositions to approximate the petrophysical parameters of an ensemble of realizations, we analyze the performance of the classical SVD and tensor approaches for low-dimensional representations of permeability fields. Let us consider an ensemble of 3D geological models with a geological structure consisting of a network of fossilized meandering channels of high perme-

144 6.1 Tensor modeling and decompositions for the representation of petrophysical properties 123 ability. The data set has been uploaded to the 4TU.Datacentrum repository and can be accessed by external users, see Jansen et al. [2014]. A view of the first layer of one of the geological realizations from the ensemble is depicted in Fig Note that here we use the full square realization with all grid blocks, whereas in the original model several grid blocks were declared void in order to modify the reservoir shape. The realizations are composed of grid blocks that are 8 8 4m in size, and have dimensions of m and 7 geological layers, and we consider R = 100 realizations. Figure 6.3: Low rank approximations using classical SVD and tensor decompositions. We construct a 4D tensor S of size , where the x, y, z and model dimensions correspond to the first, second, third and fourth tensor coordinates accordingly. Hence, the tensor S can be decomposed as in Eq. (5.17): S = i=1 j=1 z=1 r= σ ijkr ϕ i ψ j χ z ϑ r. (6.2) For instance, the permeability fields in tensor S are described by I = 60 basis functions, which are vectors ϕ R 60, for the x coordinate; J = 60 basis functions, which

145 124 Estimation of petrophysical parameters using tensor representations Figure 6.4: Relative error of the approximations. Left: Relative error as a function of the basis functions in the x and y coordinates. The color-scale corresponds to magnitude. Right: Relative error fixing the number of basis functions in one coordinate, and varying the number of basis functions in the other. are vectors ψ R 60, for the y coordinate; Z = 7 basis functions, which are vectors χ R 7, for the z coordinate; and R = 100 basis functions, which are vectors ϑ R 100, for the model coordinate. We compute low-rank approximations Ŝ of the original permeability pattern by truncating the sums in Eq. (6.2) as follows: {ϕ i }Î=20 i=1 for the x coordinate, {ψ j }Ĵ=20 j=1 for the y coordinate, {χ z }Ẑ=1 z=1 for the z coordinate and {ϑ r } R=100 r=1 for the model coordinate, leading to a modal rank approximation of (15, 35, 1, 100). The quality of the tensor approximation is compared with the classical low-rank approximation provided by the SVD, where the vector representations of the petrophysical properties are collected in a matrix S of size I J Z R, and a low-rank approximation of S is constructed by selecting a few basis functions. The visualization of one realization and the SVD and tensor approximation are depicted in Fig The results in Fig. 6.3 indicate that the rank of the SVD approximations has to be increased to 60 in order to reconstruct adequately the general patterns of the permeability fields. One relevant aspect comes from the amount of information required to approximate the rock patterns, measured in megabytes. The tensor S and the matrix S have a memory size of 20.16MB, and the basis functions and the core tensor in Eq. (6.2) for the tensor approximation of modal rank (15, 35, 1, 100) have a memory size of 0.53MB in total, which corresponds to the 2.6% of the size of the original tensor S. If we check the memory size of the basis functions and singular value matrices required to compute the low-rank SVD approximation of rank 60 of S, we find that this is 12.17MB, which corresponds to the 60.3% of the memory size of the

146 6.2 Tensor parameterizations of permeability fields 125 original matrix S. This result demonstrates the compression capabilities of tensor decompositions. This characteristic will be used in the next sections to find loworder parameterizations of permeability fields. With tensor decompositions, we are capable to approximate more accurately the patterns on a particular direction. For the case presented in Fig. 6.3, a selection of fewer basis functions on the x-coordinate filters out high order modes, producing a smoother approximation. The surface plot of the Fig. 6.4 depicts the relative error as defined in Eq. (2.64) as a function of the number of spatial basis functions in the x and y coordinates. From the surface plot it is possible to visualize that increasing the number of basis functions for the y coordinate has a bigger impact on the quality of the approximation than increasing the number of basis functions in the x coordinate. This result has a physical interpretation since the channel orientation is along the y coordinate. To confirm the directional relevance of the basis functions in the coordinate that is aligned with the channels, we have depicted in Fig. 6.4 the relative error when fixing the number of basis functions to 60 for the x or y coordinates and varying the number of basis functions for the other coordinate. The plot confirms the observation and suggest that the basis functions in the y coordinate are more relevant to approximate the features of the channels aligned with the y coordinate. For instance, tensor decompositions can be used for directional approximations that emphasize geological features like channels which are aligned with a particular spatial coordinate, and provide a framework to approximate petrophysical parameters in a more efficient way than classical SVD methods. Methods like the DCT parameterizations, see e.g. Jafarpour et al. [2009], use real cosine functions to define infinite series expansions of the permeability fields, while assuming that the most relevant geological features are preserved in the low frequency modes. Despite that this method is computationally efficient due to the fast algorithms for Fourier transforms, the approximation error is minimal if and only if the number of basis functions (cosines) tends to infinity, and in theory, it usually requires a large number of Fourier modes to capture relevant geological features. In practice, DCT-based approximations have a good performance while using a few set of basis functions, but no error estimates can be derived from the approximations. The tensor approach to the approximation of permeability fields computes the optimal basis functions which guarantee the best approximation with the computed basis functions, and error estimates can be easily computed using Eq. (2.65). In addition, it provides a framework for approximating geological features which are aligned with one specific spatial coordinate. 6.2 Tensor parameterizations of permeability fields Previously, we have exemplified the benefits of implementing tensor representations and decompositions of petrophysical parameters to extract the most relevant geological features in low-dimensional spaces. In this section, we propose a lowdimensional parameterization of the petrophysical parameters based on the tensor

147 126 Estimation of petrophysical parameters using tensor representations decomposition described in Eq. (5.17), which can be incorporated in parameter estimation routines. Then, the w th realization θ w of the ensemble of petrophysical properties in tensor form can be recovered using the induced multilinear mapping as in Eq. (2.62): I J R I J R θ w = S(,, e (w) 3 ) = σ ijr ϑ r, e (w) 3 ϕ i ψ j = αijϕ w i ψ j, i=1 j=1 r=1 i=1 j=1 r=1 (6.3) where αij w := R r=1 σ ijr ϑ r, e (w) 3 are real-valued coefficients in the expansion (6.3) of (rank 1) two-dimensional petrophysical parameter fields. The coefficient αij w is a linear combination of the w th element of the basis functions in the set {ϑ r } R r=1, i.e., αij w = R r=1 σ ijk ϑ (w) r, where ϑ (w) r = ϑ r, e (w) 3. To illustrate, suppose now that the Tucker tensor decomposition is applied to the data corresponding to an ensemble of two-dimensional rectangular permeability fields. The permeability field of the w th realization θ w is then represented as a matrix of dimension I J. A number of R realizations of petrophysical parameters θ w is stored in an order-3 tensor S of size I J R. This tensor is approximated as in Eq. (2.63), and results in the approximate sample θ w of the permeability field which can be recovered using the induced multilinear mapping as in Eq. (2.62): Î Ĵ θ w = Ŝ(,, e 3 (w) ) = R σ ijr ϑ r, e 3 (w) ϕ i ψ j = i=1 j=1 r=1 i=1 j=1 Î Ĵ αijϕ w i ψ j (6.4) where e (w) 3 is the w th standard unit vector in R R and where αij w := R r=1 σ ijr ϑ r, e (w) 3 are real-valued coefficients in the expansion (6.3) of (rank 1) two-dimensional petrophysical parameter fields. The coefficient αij w is a linear combination of the w th element of the basis functions in the set {ϑ r } R r=1, i.e., αij w = R r=1 σ ijk ϑ (w) r, where ϑ (w) r = ϑ r, e (w) 3. The expression in Eq. (2.66) is a tensor representation of the permeability fields, where all realizations of the ensemble share the same spatial basis functions defined by ϕ i ψ j. In this tensor form, the realizations are characterized by the set of coefficients α ij. This representation induces a parameterization of the ensemble if we allow the coefficients α ij to be free parameters. In the next section, we explore the use of this induced parameterization for parameter estimation.

148 6.3 EnKF for parameter estimation using tensor parameterizations EnKF for parameter estimation using tensor parameterizations The objective of CAHM is to update the model parameters through assimilating the information contained in measurements taken during the operation of the oil reservoir, which typically are the production rates and bottom-hole or tubing head pressures at the well locations. Gildin and Afra [2014] have introduced an Alternating Least Squares type of decomposition, used for the inference of permeability fields. In this section, we incorporate the tensor parameterization described in the previous section into the EnKF scheme. Let us consider the reservoir model: x k+1 = f(x k, u k, θ), (6.5) where f is a vector-valued nonlinear function of the states x k, the petrophysical parameters θ and the inputs u k. Let d sim k+1 = h(x k, u k ) be the nonlinear output operator. Using the representation in Eq. (2.66), the petrophysical parameters can be approximated by θ = Î Ĵ i=1 j=1 α ijϕ i ψ j, and can be characterized by the set of parameters α ij. Let all the parameters α ij be collected in the vector a RÎ Ĵ, and let us consider the augmented state vector z k : with augmented output vector y k : x k+1 z k+1 = a = f(z k, u k ) = f(x k, u k, θ) a, (6.6) d sim k+1 h(x k, u k ) y k+1 = Hz k+1 = [ 0 0 I ] f(x k, u k, θ) a. (6.7) h(x k, u k ) The EnKF approach follows the sequential steps of the classical KF, but uses an ensemble of R parameterizations of the realizations Ω = {a 1, a 2,, a R } to update the mean and covariance of the augmented state z: 1. Forecast: For every model realization in the set Ω, the forecast of the ensemble of augmented states is generated, and the matrix Ξ f k+1 = [ z 1,f k+1, z2,f k+1,, ] zr,f k+1 is constructed according to: z i,f k+1 = x i,f k+1 a i,f k+1 d sim,i,f k+1 = f(x i,a k, u k, i,a θ k ) a i,a k h(x i,a k, u k), (6.8)

149 128 Estimation of petrophysical parameters using tensor representations 2. Assimilation: The covariance matrix associated with the ensemble of augmented states is computed as: ( Ξ f C f k+1 = k+1 z f k+1 1 )( Ξ f k+1 zf k+1 1 ), (6.9) R 1 R i=1 zi,f where z f k+1 = 1 R k+1 is the average of the augmented states of the ensemble and 1 is the column vector with all the elements equal to 1. The Kalman gain then is defined as K k+1 = C f k+1 H (HC f k+1 H + R k ) 1, with R k the covariance matrix of the measurement noise. The states z k are assimilated with the data d k using the Kalman equations as follows: z i,a k+1 = zi,f k+1 + K k+1(d k Hz i,f k+1 ). (6.10) In practice, the EnKF is implemented in a computationally more efficient fashion. Moreover it is often necessary to implement localization techniques to avoid spurious correlations. For details we refer to Evensen [2009] and Aanonsen et al. [2009]. Figure 6.5: A realization of the permeability field and well configuration of the reservoir models. Red: Producer wells. Black: Injector wells. 6.4 Application case In this section we perform a parameter estimation experiment where we compare the performance of the tensor parameterizations described in this chapter with classical

150 6.4 Application case 129 Parameter Value Unit Simulation parameters Simulation time 15 [year] Time step 1 [month] Measurement errors, standard deviations Flow rates 12 [m 3 /day] Bottom-hole pressures 6 [bar] Fractional flow (watercuts) Parameter estimation experiment Observation time interval 3 [month] Update time interval 2 [year] End of history matching 6 [year] Table 6.1: Details of the experimental setting. PCA parameterizations. We consider the data set provided by Vo and Durlofsky [2015], and construct an ensemble of R = 100 realizations of channelized reservoirs of size I = 60 and J = 60. The well configuration consists of 12 producer wells and 4 injectors, and their locations are depicted in Fig The producers are operated at a fixed bottom hole pressure and the injectors at fixed rates. The field is producing for 15 years, with a simulation time step of t = 1 month. The measurements to be considered are the total rates at the producers, the bottomhole pressures at the injectors and the fractional flows at the producers, with an observation time interval of 3 months. In the experiment, measurement errors are modeled as white noise, i.e. stochastic variables with a Gaussian distribution of zero mean. During the production stage, the petrophysical parameters are estimated with an interval of 2 years and the last update is made 6 years after the start of production. The key parameters for the experiment are summarized in Table 6.1. The forward simulations are performed with the Matlab Reservoir Simulation Toolbox (MRST), see, e.g., Lie et al. [2012]. For the parameter estimation, we have used the EnKF module of MRST. We choose realization number 100 to be the truth. The permeability fields are represented in a tensor formulation S of size and we construct a set of low-dimensional representations Ω = {a 1, a 2,, a 99 }, where a R 50, which is used as prior ensemble for the EnKF routine. The plots of the total flow rates in several production wells, after the EnKF final update, are presented in Figs. (6.7)-(6.9). All of the plots consist of three figures, each one associated with the flow rates in a particular well for the parameter estimation result when using the unparameterized EnKF, the EnKF with tensor parameterization and the EnKF with PCA parameterization. The history matching period, which is depicted with a green background, ends after 6 years, and thereafter the reservoir models are used for prediction, during the time period depicted with a pink background. One of the principal objectives of the parameter estimation of reservoir models, of more importance than having a good history-match of the reservoir out-

151 130 Estimation of petrophysical parameters using tensor representations Figure 6.6: History matching and rate prediction. Total rates at well 5. The green background indicates the history matching interval. The pink background indicates the prediction interval. Red: Truth. Gray: Updated ensemble. puts, is to create an updated ensemble of reservoir models with good predictive capabilities. On the one hand, the EnKF without parameterization shows a very good history matching result. However, the updated ensemble of realizations presents very poor prediction capabilities for most of the wells. On the other hand, when the EnKF is used with low-dimensional tensor and the PCA parameterizations, it still generates good history matching results but now also results in considerably improved prediction capabilities. The poor prediction capability of the EnKF without parameterization can be explained from the loss of geological realism of the updated ensemble. In Fig and Fig we depict the final permeability updates of two realizations, using the methods described in this section. When no parameterization is used, the updates totally lose the channel structure. This is because the Kalman filter is only capable to capture the second-order statistics of the petrophysical properties. When tensor parameterizations are used, combined with an EnKF, they impose a certain amount of geological structure on the updates. Despite that the EnKF still only captures the second-order statistics, the geological updates have a structure imposed by the prior knowledge, which is included in the tensor basis functions, and as a result the geological updates still look like channelized reservoir models. The PCA updates lose much more of the geological realism of channelized structures compared to the results from the tensor updates.

152 6.4 Application case 131 Figure 6.7: History matching and rate prediction. Total rates at well 5. The green background indicates the history matching interval. The pink background indicates the prediction interval. Red: Truth. Gray: Updated ensemble. Similar observations have been found for the watercuts. To exemplify, Fig and Fig show that the watercuts predicted by the EnKF with no parameterization are deviating from the truth, while the watercuts from the updated ensemble using a low-dimensional representation are able to predict more accurately the water breakthrough times.

153 132 Estimation of petrophysical parameters using tensor representations Figure 6.8: History matching and rate prediction. Total rates at well 11. The green background indicates the history matching interval. The pink background indicates the prediction interval. Red: Truth. Gray: Updated ensemble.

154 6.4 Application case 133 Figure 6.9: History matching and rate prediction. Total rates at well 15. The green background indicates the history matching interval. The pink background indicates the prediction interval. Red: Truth. Gray: Updated ensemble. Figure 6.10: Final update permeability field, realization 78.

155 134 Estimation of petrophysical parameters using tensor representations Figure 6.11: Final update permeability field, realization 75. Figure 6.12: History matching and water breakthrough prediction. Fractional flow at well 5. The green background indicates the history matching interval. The pink background indicates the prediction interval. Red: Truth. Gray: Updated ensemble.

156 6.4 Application case 135 Figure 6.13: History matching and water breakthrough prediction. Fractional flow at well 9. The green background indicates the history matching interval. The pink background indicates the prediction interval. Red: Truth. Gray: Updated ensemble.