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  • PHDGProfessor Heinemanns

    DoktorandengruppeVerein zur Frderung

    von wissenschaftlichen Arbeitenin Reservoircharakterisierung und -simulation

    TEXTBOOK SERIES

    VOLUME 1

    FLUID FLOW IN POROUS MEDIA

    by

    Zoltn E. HEINEMANNProfessor for Reservoir Engineering

    Leoben, October 2005

    actualized by

    Dr. Georg MittermeirTehran, February 2013

  • Roseggerstr. 11a | 8700 Leoben | Austria | Phone: +43 (0)3842/4331611 | E-Mail: [email protected]: AT78 1200 0100 0531 8976 | BIC: BKAUATWW

    For kind Attention

    PHDGs Textbooks available at 1.1.2015:

    1. Fluid Flow in Porous Medium2. Well Testing3. Systematic of the Reservoir Flow Equations4. Introduction to Reservoir Simulation5. Natural Fractured Reservoir Engineering

    PHDG Textbooks in preparation, intended to be issued during 2015:

    1. Discretization and Gridding in Reservoir Simulation2. Advanced Reservoir Simulation3. Reservoir Fluid Characterisation

    Supplementary scripts used at the Montanuniversitt up to the retirement of ProfessorZoltn E. Heinemann in July 2006.

    1. Reservoir Fluids2. Petroleum Recovery

    No part of this publication may be reproduced in any form.Not applicable as teaching material at universities or any other kind of courses without prior, writtenpermission of the PHDG association. Students of the following universities can ask for free copies forpersonal use: Sharif University of Technology, Tehran University, Iran University of Science andTechnology, Shiraz University, University of Miskolc, Montanuniversitt Leoben.

  • Table of Contents 3

    Table of Contents1 Fundamental Properties of Porous Media ................................................... 5

    1.1 Porosity ....................................................................................................................... 61.1.1 General Aspects and Definition ........................................................................ 61.1.2 Determination of Porosity ................................................................................. 71.1.3 Compaction ....................................................................................................... 91.1.4 Compressibility of Porous Media ..................................................................... 91.1.5 Classification of Porosity ................................................................................ 12

    1.2 Capillary Properties ................................................................................................... 141.2.1 Saturation ........................................................................................................ 141.2.2 Wettability ...................................................................................................... 14

    1.2.2.1 Classification of Reservoir Rocks based on Wettability .................... 151.2.2.2 Measurement of Wettability ............................................................... 16

    1.2.2.2.1 Contact Angle Measurement ......................................... 171.2.2.2.2 Amott Method1.19.......................................................... 181.2.2.2.3 USBM Wettability Index............................................... 19

    1.2.3 Capillary Pressure ........................................................................................... 231.2.3.1 Definition............................................................................................ 231.2.3.2 Measurement of Capillary Pressure in a Porous Medium.................. 25

    1.2.3.2.1 Method of Centrifuge .................................................... 271.2.3.2.2 Mercury Injection (Purcell Method).............................. 31

    1.2.3.3 Conversion of Laboratory Data .......................................................... 321.2.4 The Leverett Function ..................................................................................... 341.2.5 Pore Size Distribution ..................................................................................... 351.2.6 Vertical Equilibrium ....................................................................................... 36

    1.3 Permeability .............................................................................................................. 391.3.1 Darcys Law ................................................................................................... 391.3.2 Definition and Units of Permeability .............................................................. 401.3.3 Measurements of Permeability ....................................................................... 421.3.4 Klinkenberg Effect .......................................................................................... 461.3.5 Analogies between the Laws of Darcy, Ohm and Fourier ............................. 471.3.6 Filtration Velocity ........................................................................................... 481.3.7 Quadratic Equation of Filtration ..................................................................... 49

    1.4 Relative Permeabilities ............................................................................................. 501.4.1 Definition of Relative Permeability ................................................................ 511.4.2 Definitions of End-Point Saturations .............................................................. 511.4.3 Relative Permeability Measurements ............................................................. 53

    1.4.3.1 The HASSLER method ...................................................................... 531.4.3.2 PENN-STATE-Method ...................................................................... 571.4.3.3 Welge-Method.................................................................................... 58

    1.4.4 Saturation Distribution and Relative Permeability ......................................... 581.5 References ................................................................................................................. 61

    2 Equations of Single-Phase Filtration .......................................................... 65

  • Table of Contents 4

    2.1 Fundamental Equation of Filtration. ......................................................................... 672.1.1 Differential Form of the Darcy-Law .............................................................. 672.1.2 Anisotropic Porous Media .............................................................................. 70

    2.2 Equation of State ....................................................................................................... 732.2.1 Incompressible Fluids ..................................................................................... 732.2.2 Low Compressibility Fluids ........................................................................... 732.2.3 Formation Volume Factor ............................................................................... 742.2.4 Ideal and Real Gases ....................................................................................... 752.2.5 Equation of continuity .................................................................................... 76

    2.3 Special Forms of the Equation of Filtration .............................................................. 792.3.1 Incompressible Fluids ..................................................................................... 792.3.2 Low Compressibility Fluids ........................................................................... 79

    2.3.2.1 Elastic Porous Media.......................................................................... 802.4 Real and Ideal Gases ................................................................................................. 822.5 Boundary and Initial Conditions ............................................................................... 84

    2.5.1 Boundary Conditions ...................................................................................... 842.5.2 Initial Conditions ............................................................................................ 852.5.3 Discontinuities in Porous Media ..................................................................... 86

    2.6 Schematic of the Filtration Equations ....................................................................... 87

    3 Solutions of the Single-Phase Equation of Filtration ................................ 913.1 Steady State Filtration ............................................................................................... 92

    3.1.1 Steady State Filtration of Low Compressibility Fluid .................................... 923.1.2 Steady State Filtration in a Radial System ..................................................... 933.1.3 Steady State Gas Filtration ............................................................................. 96

    3.2 Non-Steady State Filtration in Infinite Acting Systems ............................................ 983.2.1 Radial Systems with Constant Production Rate ............................................. 983.2.2 Properties of the Ei-Function ........................................................................ 1023.2.3 Pressure Drop in Space and Time ................................................................. 1033.2.4 The Spatial Distribution of Flow .................................................................. 107

    3.3 Dimensionless Variables ......................................................................................... 1083.4 The Infinite Radial System with Constant Pressure at the Interior Boundary ........ 1133.5 Non-Steady State Filtration in a Finite System ....................................................... 118

    3.5.1 Constant Production Rate