Copyright by Alvaro Enrique Barrera 2007 The Dissertation Committee for Alvaro Enrique Barrera certifies that this is the approved version of the following dissertation: History Matching by Simultaneous Calibration of Flow Functions Committee: _________________________________ Sanjay Srinivasan, Supervisor _________________________________ Steven L. Bryant _________________________________ A. Stan Cullick _________________________________ Larry W. Lake _________________________________ Gary A. Pope History Matching by Simultaneous Calibration of Flow Functions by Alvaro Enrique Barrera, B.S.; M.S. Dissertation Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy The University of Texas at Austin December 2007 Acknowledgements I would like to express sincere gratitude to my supervisor and mentor, Dr. Sanjay Srinivasan. Dr. Srinivasan’s expert advice and creative insights have had a profound and undeniable impact upon this dissertation. More importantly, my career and personal life have benefited greatly from Dr. Srinivasan’s ongoing support and honest guidance during my graduate studies. I would also like to thank Steven Bryant, Stan Cullick, Larry Lake and Gary Pope for their commitment of time, as well as their constructive observations, while serving on my dissertation committee. I would also like to thank these gentlemen for their scholastic contributions not only to this dissertation but also to my future career in Petroleum Engineering. To my fellow graduate research assistants, Shinta Reinlie, Juliana Leung, Kiomars Eskandari, Yonghwee Kim, Louis Forster, Cesar Mantilla and Aviral Sharma, I thank you for your unfailing assistance and loyal friendship during my studies at The University of Texas at Austin. iv History Matching by Simultaneous Calibration of Flow Functions Publication No._____________ Alvaro Enrique Barrera, Ph.D. The University of Texas at Austin, 2007 Supervisor: Sanjay Srinivasan Reliable predictions of reservoir flow response corresponding to various recovery schemes require a realistic geological model of heterogeneity and an understanding of its relationship with the flow properties. This dissertation presents results on the implementation of a novel approach for the integration of dynamic data into reservoir models that combines stochastic techniques for simultaneous calibration of geological models and multiphase flow functions associated with pore- level spatial representations of porous media. In this probabilistic approach, a stochastic simulator is used to model the spatial distribution of a discrete number of rock types identified by rock/connectivity indexes (CIs). Each CI corresponds to a particular pore network structure with a v characteristic connectivity. Primary drainage and imbibition displacement processes are modeled on the 3-D pore networks to generate multiphase flow functions corresponding to networks with different CIs. During history matching, the stochastic simulator perturbs the spatial distribution of the CIs to match the simulated pressures and flow rates to historic data, while preserving the geological model of heterogeneity. This goal is accomplished by applying a probabilistic approach for gradual deformation of spatial distribution of rock types characterized by different CIs. Perturbation of the CIs in turn results in the update of all the flow functions including the effective permeability, porosity of the rock, the relative permeabilities and capillary pressure. The convergence rate of the proposed method is comparable to other current techniques with the distinction of enabling consistent updates to all the flow functions. The resultant models are geologically consistent in terms of all the flow functions, and consequently, predictions obtained using these models are likely to be more accurate. To compare and contrast this comprehensive approach to reservoir modeling against other approaches that rely on modeling and perturbing only the permeability field, a realistic case study is presented with implementation of both approaches. Comparison is made with the history-matched model obtained only by perturbing permeability. It is argued that reliable predictions of future production can only be made when the entire suite of flow functions is consistent with the real reservoir. vi Table of Contents List of Tables ……………………………………………………………………....... x List of Figures ……………………………………………………………...….......... xi 1 INTRODUCTION ..................................................................................................... 1 1.1 Problem Statement.............................................................................................. 2 1.2 Objectives ........................................................................................................... 3 1.3 Approach Overview............................................................................................ 4 1.4 dissertation outline.............................................................................................. 6 2 REVIEW OF RELEVANT LITERATURE .............................................................. 8 2.1 Dynamic Data Integration................................................................................... 8 2.1.1 Calibration of the Static Model.................................................................. 10 2.1.2 Calibration of Multiphase Flow Functions ................................................ 14 2.2 Pore Space Structure......................................................................................... 16 2.3 Pore Network Models ....................................................................................... 21 3 MODELING FRAMEWORK ................................................................................. 25 3.1 Workflow .......................................................................................................... 25 3.2 Pore scale representation of real lithology........................................................ 26 3.3 Scaling up pore scale flow functions to simulation inputs ............................... 29 3.4 Parameters for History Matching...................................................................... 31 3.5 Assessing the accuracy of results...................................................................... 33 4 PROBABILISTIC HISTORY MATCHING........................................................... 36 4.1 Field-Scale Stochastic Modeling ...................................................................... 36 4.2 Sequential Indicator Simulation........................................................................ 39 4.3 Dynamic data assimilation................................................................................ 44 4.3.1 Merging information from dynamic and static sources ............................. 45 4.3.2 Gradual Calibration of Reservoir Models.................................................. 49 vii 4.3.2.1 Original Perturbation Scheme......................................................... 50 4.3.2.2 Second perturbation scheme ........................................................... 56 4.3.2.3 Third perturbation scheme.............................................................. 57 4.3.2.4 Final perturbation scheme............................................................... 59 4.3.3 Deformation parameter optimization......................................................... 62 4.3.4 Objective Function..................................................................................... 65 4.4 Gradual updating procedure for history matching............................................ 66 4.5 Implementation of the Method.......................................................................... 69 4.6 Preliminary Validation...................................................................................... 72 5 PORE-LEVEL REPRESENTATIONS ................................................................... 77 5.1 Pore space description....................................................................................... 78 5.2 Pore Network Model......................................................................................... 82 5.3 Pore network simulator..................................................................................... 88 5.3.1 Primary Drainage....................................................................................... 90 5.3.2 Imbibition................................................................................................... 91 5.3.3 Capillary Pressure Curves.......................................................................... 93 5.3.4 Relative Permeability Curves. ................................................................... 94 6 CALIBRATION OF NETWORK RESULTS......................................................... 98 6.1 Effect of Pore Network Model Size.................................................................. 99 6.2 Effect of Spatial Correlation ........................................................................... 101 6.3 Porosity Effect ................................................................................................ 107 6.4 Effect of Grain Size Sorting............................................................................ 108 6.5 Effect of Layers............................................................................................... 110 6.5.1 Horizontal Layers....................................................................................