Techniques for Modeling Complex Reservoirs and Advanced Wells

TECHNIQUES FOR MODELING COMPLEX RESERVOIRS AND ADVANCED WELLS A DISSERTATION SUBMITTED TO THE DEPARTMENT OF ENERGY RESOURCES ENGINEERING AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Yuanlin Jiang December 2007 °c Copyright by Yuanlin Jiang 2008 All Rights Reserved ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Dr. Hamdi Tchelepi Principal Advisor I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Dr. Khalid Aziz Advisor I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Dr. Roland Horne Approved for the University Committee on Graduate Studies. iii Abstract The development of a general-purpose reservoir simulation framework for coupled systems of unstructured reservoir models and advanced wells is the subject of this dissertation. Stanford's General Purpose Research Simulator (GPRS) serves as the base for the new framework. In this work, we made signi¯cant contributions to GPRS, in terms of architectural design, extensibility, computational e±ciency, and new advanced well modeling capabilities. We designed and implemented a new architectural framework, in which the facilities (man-made) model is treated as a separate component and promoted to the same level as the reservoir (natural) component. The framework is quite general and extensible. It reduces the cost of incorporating new models into GPRS signi¯cantly. This is demonstrated by incorporating several new facility capabilities, namely, the Multisegment Well (MSWell) model and handling well-group constraints. Two new data structures have been built for GPRS to accommodate robust and computation- ally e±cient simulation. They honor the separation of the reservoir and facilities components and give researchers freedom to develop their own discrete formulations and solution methods. The new data structures also simplify the construction and assembly of Jacobian matrices. The MSWell was fully integrated into the new GPRS framework. This can be seen iv as an example of a signi¯cant capability extension of a general-purpose simulator. Moreover, a new rigorous well-group treatment is proposed and implemented. This new model accounts for interactions between the wells in the group and allows for using one's favorite pipeline flow correlation. The well-group model bene¯ts from the flexible data structures of GPRS. For example, one can combine any available facilities object such as standard and multisegment wells and mixed well grouping of both kinds. With these new extensions, GPRS can simulate coupled systems of (unstructured) reservoir models and advanced wells. A general discrete wellbore model (GenWell) that extends the MSWell treatment from the perforation all the way to the surface was developed. The GenWell model shares the advantages of the MSWell model, but it also accommodates complex pipeline topology, such as general branching, loops and multiple exits. In the GenWell model, wellbores and pipeline networks are discretized into various types of segments. Nodes and connections are de¯ned based on the segments, and the discrete system is abstracted as a graph (nodes and connections) in a manner that is quite similar to the representation of unstructured reservoir models in GPRS. As part of this work, advanced multistage linear solution strategies were developed. Based on the new block data structures, we developed and implemented block ILU preconditioners. These block preconditioners help further speed up the simulation. The two-stage CPR (Constrained Pressure Residual) preconditioning approach is extended to handle systems with the MSWell model. The data structures and linear solvers we developed are fully compatible with AIM (Adaptive Implicit Method). We demonstrate that the CPR preconditioned GMRES solver based on the new framework is the best linear solution strategy for large-scale AIM simulation of unstructured reservoir models and advanced wells. v Acknowledgement I would like to express my sincere thanks to my advisers, Prof. Hamdi Tchelepi, and Professor Khalid Aziz for their support, encouragement and guidance during this work. Prof. Hamdi Tchelepi has always been deeply involved with the work and spent tremendous time on it. Prof. Khalid Aziz was my M.S. adviser and is my Ph.D. co-adviser. He has been a consistent source of knowledge, advice and support, since the ¯rst day I got to Stanford. I would like to thank Prof. Roland Horne for reading my thesis and providing valuable suggestions and corrections. I also bene¯ted from his contribution to the department as Chairman. Prof. Eric Darve kindly chaired my Ph.D. oral defense and is gratefully acknowledged. I would like to thank Prof. Louis Durlofsky for being my defense committee member. His valuable comments are highly appreciated. I worked with Dr. Hua Shi and Jeremie Batias to integrate the MSWell model into GPRS. Their contributions are acknowledged. I also would like to thank Dr Huanquan Pan, who is the administrator of GPRS source code, for his support. I would like to express my greatest gratitude to the Department of Energy Re- sources Engineering at Stanford University, all its faculty, sta® and students. I feel lucky for being able to spend ¯ve years in this inspiring and friendly environment. vi The friendships with them are the most important treasure of my life. I would like to thank the industrial a±liates of the Stanford University Reservoir Simulation (SUPRI-B) and Advanced Wells (SUPRI-HW) research programs. Their ¯nancial supports made this work possible and are gratefully acknowledged. Finally, I would like to thank my wife Weimin Sun for her unconditional support and for giving me the joy of life, our newborn son, Frank. I would also like to thank my mother-in-law for helping us take care of Frank. My parents and sister in China also deserve my gratitude for supporting and encouraging my education from the very beginning. vii Contents Abstract iv Acknowledgement vi 1 Introduction and Objectives 1 1.1 Background ................................ 1 1.1.1 General-Purpose Reservoir Simulation ............. 3 1.1.2 Discrete wellbore models ..................... 6 1.1.3 Coupled Reservoir-Facilities Simulation ............. 7 1.2 Objectives ................................. 9 1.3 Dissertation Outline ........................... 10 2 Framework for Reservoir-Facilities Simulation 12 2.1 New GPRS Framework .......................... 14 2.2 `SimMaster' and Reservoir Classes .................... 16 viii 2.3 `Facilities' Class .............................. 16 2.4 Interface of Key Classes ......................... 18 3 Innovative Data Structures in GPRS 20 3.1 Original Data Structures in GPRS ................... 22 3.1.1 Raw Data ............................. 22 3.1.2 Compressed Row Sparse Matrix Format ............ 24 3.2 Multilevel Sparse Block Matrix ..................... 27 3.2.1 First Level: Global Matrix .................... 27 3.2.2 Second Level: Reservoir and Facilities Matrices ........ 31 3.2.3 Third Level: Well Matrices .................... 34 3.2.4 Matrix-Vector Multiplication ................... 34 3.3 Block Compressed Row Sparse Matrix ................. 36 3.4 Concluding Remarks ........................... 40 4 Well Modeling 41 4.1 Standard Well Model ........................... 42 4.1.1 Variables and Equations ..................... 42 4.1.2 Density Treatment ........................ 44 4.2 Multisegment Well (MSWell) Model ................... 46 ix 4.2.1 Geometry and Variables ..................... 46 4.2.2 Momentum and Mass Balance Equations ............ 50 4.2.3 Constraint Equations and Jacobian Matrix Structures ..... 53 4.2.4 MSWell Initialization ....................... 62 4.2.5 Homogeneous and Drift-flux Models ............... 65 4.2.6 Test Cases ............................. 68 4.3 Concluding Remarks ........................... 77 5 Well Group Modeling 79 5.1 Objects and Concepts .......................... 80 5.2 Jacobian Matrix Structure ........................ 83 5.3 Governing Equations and Variables ................... 85 5.4 Example .................................. 88 5.5 Concluding Remarks ........................... 89 6 General Pipeline and Wellbore Modeling 92 6.1 Segments, Nodes and Connections .................... 93 6.2 Governing Equations and Variables ................... 96 6.3 Example .................................. 99 6.4 Comments on Future Extension ..................... 104 x 7 Linear Solution Strategies 107 7.1 Linear Solvers and Preconditioners in GPRS .............. 108 7.2 Introduction to GMRES ......................... 111 7.3 Single-stage Preconditioners ....................... 114 7.3.1 Introduction ............................ 114 7.3.2 Incomplete LU Factorization (ILU) ............... 117 7.3.3 Algebraic Multigrid (AMG) ................... 124 7.4 CPR for Reservoir Models with StdWells ................ 125 7.4.1 First Stage: Pressure System .................. 126 7.4.2 Second Stage: Simple Preconditioning of the Full System ... 132 7.4.3 Test Cases ............................. 133 7.5 CPR for Reservoir Models with Multisegment Wells .......... 144 7.5.1 First Stage: Pressure Equation

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