An Introduction to Multiphase, Multicomponent Reservoir Simulation

An Introduction to Multiphase, Multicomponent Reservoir Simulation

1st Edition - September 1, 2022

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  • Author: Matthew Balhoff
  • Paperback ISBN: 9780323992350

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Description

An Introduction to Petroleum Reservoir Simulation is aimed toward graduate students and professionals in the oil and gas industry working in reservoir simulation. It begins with a review of fluid and rock properties and derivation of basic reservoir engineering mass balance equations. Then equations and approaches for numerical reservoir simulation are introduced. The text starts with simple problems (1D, single phase flow in homogeneous reservoirs with constant rate wells) and subsequent chapters slowly add complexities (heterogeneities, nonlinearities, multi-dimensions, multiphase flow, and multicomponent flow). Partial differential equations and finite differences are then introduced but it will be shown that algebraic mass balances can also be written directly on discrete grid blocks that result in the same equations. Many completed examples and figures will be included to improve understanding. An Introduction to Petroleum Reservoir Simulation is designed for those with their first exposure to reservoir simulation, including graduate students in their first simulation course and working professionals who are using reservoir simulators and want to learn more about the basics.

Key Features

  • Presents basic equations and discretization for multiphase, multicomponent transport in subsurface media in a simple, easy-to-understand manner
  • Features illustrations that explain basic concepts and show comparison to analytical solutions and commercial simulators
  • Includes dozens of completed example problems on a small number of grid blocks
  • Offers pseudocode and exercises to allow the reader to develop their own computer-based numerical simulator that can be verified against analytical solutions and commercial simulators

Readership

Graduate students and professionals in the oil and gas working in reservoir simulation. Upper-level undergraduate students in their first simulation course and geologists working in the oil and gas industry. Applicable courses or exams: Undergraduate and graduate courses in reservoir simulation (in petroleum geology and engineering programs)

Table of Contents

  • 1. Review of Rock and Fluid Properties
    1.1 Introduction  
    1.2 Definitions
    1.2.1 Phases and Components in Subsurface Porous Media
    1.2.2 Density, Saturation, and Concentrations
    1.3 Phase Behavior
    1.4 Rock and Fluid Properties
    1.4.1 Formation properties
    1.4.2 Gaseous phase properties
    1.4.3 Oleic phase properties
    1.4.4 Aqueous phase properties
    1.5 Petrophysical properties
    1.5.1 Darcy’s Law
    1.5.2 Relative Permeability
    1.5.3 Capillary Pressure  
    1.6 Reservoir Initialization  
    References  

    Chapter 2. Single Phase Flow in Porous Media  
    2.1 Introduction 
    2.2 Continuity Equation  
    2.2.1 Cartesian coordinates  
    2.2.2 Cylindrical coordinates  
    2.2.3 General form  
    2.3 Specific cases of single phase flow  
    2.3.1 Flow of a single aqueous phase  
    2.3.2 Flow of a single oleic phase in the presence of immobile water  
    2.3.3 Flow of a single gaseous phase in the presence of immobile water (primary recovery)  
    2.4 Diffusivity equation  
    2.4.1 Slightly compressible flow  
    2.4.2 Compressible flow  
    2.4.3 Cylindrical Coordinates  
    2.5 analytical solutions  
    2.5.1 1D Heat Equation in Finite Medium  
    2.5.2 1D Heat equation in a semi-infinite medium  
    2.5.3 1D Diffusivity equation in cylindrical coordinates (radial flow in infinite-acting and finite domains)  
    2.5.4 Numerical solutions to the diffusivity equation  
    References  

    Chapter 3. Finite Difference Solutions to the 1D Diffusivity Equation  
    3.1 Introduction  
    3.2 Taylor Series and Finite Differences  
    3.2.1 First-Order Forward Difference Approximation  
    3.2.2 First-Order Backward Difference Approximation  
    3.2.3 Second-Order Centered Difference Approximation  
    3.2.4 Approximations to the Second Derivative  
    3.2.5 Higher-Order Approximations  
    3.3 Discretization of the Parabolic Diffusivity (Heat) Equation  
    3.3.1 Explicit Solution to the Diffusivity Equation  
    3.3.2 Implicit Solution to the Diffusivity Equation  
    3.4 Boundary and Initial Conditions  
    3.4.1 Dirichlet Boundary Condition  
    3.4.2 Neumann Boundary Condition  
    3.4.3 Other Boundary Conditions  
    3.5 Stability and Convergence  
    3.6 Matrix Equations in Terms of Flow Units  
    3.7 Mixed Methods and Crank-Nicholson  
    3.8 Pseudocode  
    3.9 Point Distributed Grids  
    References  

    Chapter 4. Control Volume Approach, Heterogeneities, Gravity, and Nonlinearities  
    4.1 Introduction  
    4.2 Control Volume Approach  
    4.2.1 Flow in and out of blocks  
    4.2.2 Accumulation  
    4.2.3 Sources and Sinks (Wells)  
    4.2.4 Balance Equations  
    4.3 Reservoir heterogeneities  
    4.3.1 Permeability  
    4.3.2 Variable grid blocks  
    4.3.3 Other Heterogeneities  
    4.3.4 Mass Balance Equations  
    4.3.5 Boundary Conditions  
    4.4 Gravity  
    4.5 Non-Linearities in Reservoir Simulation  
    4.5.1 Explicit Update of Fluid and Reservoir Properties  
    4.5.2 Fixed Point Iteration  
    4.5.3 Newton’s method  
    4.5.3.1 1D Newton’s method  
    4.5.3.2 Multidimensional Newton’s method  
    4.6 Pseudocode  
    4.7 Examples of Nonlinearities in Single-Phase Flow Problems  
    4.7.1 Gas Flow  
    4.7.1.3 General Numerical Solution to Gas Flow  
    4.7.2 Non-Newtonian Flow  
    4.7.3 Forchheimer Flow  
    References  

    CHAPTER 5. 2D AND 3D SINGLE-PHASE FLOW  
    5.1 Introduction  
    5.2 Grid Block Numbering in 2D  
    5.3 Mass Balance Equations  
    5.3 Special Issues in 2D Flow  
    5.3.1 Interblock Transmissibility  
    5.3.4 Irregular Geometry and Inactive Grids  
    5.4 Initial and Boundary Conditions  
    5.4.1 Neumann Conditions  
    5.5.2 Dirichlet Conditions  
    5.5.3 Corner blocks  
    5.6 3D Flow  
    5.7 Pseudocode for 2D and 3D  
    References  

    CHAPTER 6. WELLS, WELL MODELS, AND RADIAL FLOW  
    6.1 Introduction  
    6.2 Radial Flow Equations and Analytical Solutions  
    6.3 Numerical Solutions to the Radial Diffusivity Equation  
    6.3.1 Direct Discretization  
    6.3.2 Variable Transformation  
    6.3.3 2D and 3D flow problems in cylindrical coordinates  
    6.4 Wells and Well Models in Cartesian Grids  
    6.4.1 Constant Rate Wells  
    6.4.2 Constant Bottom Hole Pressure (BHP) Wells  
    6.4.2.1 Steady-state radial flow around a well  
    6.4.2.2 Mass Balance on the well-containing grid block  
    6.4.2.3 Alternatives to Peaceman’s correction  
    6.5 Inclusion of the well model into the matrix equations  
    6.6 Horizontal Wells  
    References  

    Chapter 7. Component Transport in Porous Media  
    7.1 Introduction  
    7.2 Transport Mechanisms  
    7.2.1 Advection  
    7.2.2 Diffusion and Dispersion  
    7.2.3 Reaction  
    7.3 Component Balance Equations  
    7.3.1 Cylindrical coordinates  
    7.3.2 General form  
    7.4 Analytical solutions to the ADE  
    7.5 Finite Difference Solution to the ADE  
    7.5.1  Implicit Pressure, Explicit Concentration (IMPEC)  
    7.5.2 Sequential Implicit  
    7.5.3 Fully Implicit  
    7.6 Boundary Conditions and wells  
    7.6.1 Constant concentration (Dirichlet)  
    7.6.2 Mixed (specified flux)  
    7.6.3 No flux boundary  
    7.6.4 Wells  
    7.7 Upwinding  
    7.8 Stability  
    7.9 Numerical Dispersion  
    7.10 Viscous Fingering (TBD)  
    7.11 Multicomponent Transport (TBD)  
    7.12  Reactive Transport (TBD)  
    References  

    Chapter 8. Numerical Solution of the Black Oil Model  
    8.1 Introduction  
    8.2 Derivation of 3-phase black oil equations  
    8.3 Analytical Solutions – Bulkley-Leverett (Note: Taken partially from Gary Pope’s (2008) Reservoir II notes)  
    8.4 Numerical solution to 1d, 2-phase undersaturated oil  
    8.4.1 Finite Differences (1D, 2-phase flow, no capillary pressure or gravity)  
    8.4.2 Implicit Pressure, Explicit Saturation (IMPES)  
    8.4.3 SS Method  
    8.4.4 Interblock transmissibilities and upwinding  
    8.4.5 Wells and Well Models  
    8.5 2-phase, 2D and 3D flow with capillary pressure and gravity  
    8.5.1 IMPES solution  
    8.5.2 Simultaneous Solution (SS)  
    8.5.3 Simultaneous Solution (SS) #2 Method  
    8.5.4 Fully Implicit Method  
    8.6 Numerical solutions to 3-phase flow (black oil model)  
    REFERENCES  

    Chapter 9. Multiphase Compositional Modeling  
    9.1 Introduction  
    9.2 Multiphase Flow Compositional Equations  
    9.3 Finite Difference Equations  
    9.4 Flash Calculations  
    9.4.1 K-values  
    9.4.2 Flash Calculation  
    9.5 Equations of State  
    9.5.1 Cubic EOS  
    9.5.2 Mixing Rules  
    9.5.3. Solution to the cubic EOS  
    9.5.3. Analytical solution to cubic polynomials  
    9.5.4 K-value calculation  
    9.6 Phase Viscosity  
    9.7 Phase Saturation, Relative Permeability, and Transmissibility  
    9.9 Moles and Accumulation  
    9.10 Pressure and Composition Solution  
    REFERENCES  

Product details

  • No. of pages: 200
  • Language: English
  • Copyright: © Elsevier 2022
  • Published: September 1, 2022
  • Imprint: Elsevier
  • Paperback ISBN: 9780323992350

About the Author

Matthew Balhoff

Matthew T. Balhoff is a Professor in the Hildebrand Department of Petroleum and Geosystems Engineering and Director of the Center for Subsurface Energy and the Environment at UT-Austin. He co-leads the Industrial Affiliates Program on Chemical Enhanced Oil Recovery. Dr. Balhoff received his BS (2000) and PhD (2005) in chemical engineering from Louisiana State University. He became an SPE Distinguished member in 2017 and is a winner of the 2017 SPE Southwestern North America Regional Reservoir Description and Dynamics Award, 2014 SPE International Young Member Service Award, and 2012 SPE International Teaching Fellow Award. Dr. Balhoff has over 70 peer-reviewed publications in the areas of numerical reservoir simulation, pore-scale modelling, enhanced oil recovery, carbon storage, and unconventional resource production. He has taught dozens of undergraduate and graduate courses on numerical reservoir simulation, reservoir engineering, and fluid properties.

Affiliations and Expertise

Director, Center for Subsurface Energy and the Environment; Professor, Hildebrand Department of Petroleum and Geosystems Engineering, The University of Texas, Austin, TX, USA; Bank of America Professorship in Petroleum Engineering

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