A Simulation Model
for Two-Phase Flow in
Porous Media

Eyvind Aker
Thesis for the Degree of
Candidatus Scientiarum
Department of Physics
University of Oslo
December 1996

• Contents
• Introduction
• Flow in Porous Media
  • Reservoir Flow
    • Multi Phase Flow
  • Microscopic Physics
    • Interfacial Tension
    • Wettability
    • Drainage and Imbibition
    • Capillary Flow in a Tube
  • Drainage Displacements
    • Fluid Forces
      • ``Phase-diagram'' of Lenormand et al.
    • Viscous Fingering
      • DLA
    • Stable Displacement
      • Anti-DLA
    • Capillary fingering
      • Invasion Percolation
    • Network Simulators
• The Simulation Model
  • The Motivation for the Model
  • Description of the Model
    • The Geometry
    • Flow Conditions
    • Flow in a Single Tube
      • Extension to Two Pore-Interfaces in a Single Tube
      • Units
  • Solving the Flow Field
    • The Kirchhoff Equations
      • Solving for a Constant Volume Flow Rate
    • The Analogy With a Random Resistor Network
  • Updating the Pore-Interfaces
    • Finding the Time Step
      • In the Middle of the Tube
    • The Runge-Kutta Method
    • Moving the Pore-Interfaces Into Neighbor Tubes
      • Trapped Tubes
      • Mixing of the Fluids
  • Convergence and Stability Analysis
• Results and Discussion
  • The Simulations
    • Capillary Number and Viscosity Ratio
  • The Movie of the Three Typical Regimes
    • The Regime of Viscous Fingering
    • The Regime of Stable Displacements
    • The Regime of Capillary Fingering
    • Numerical comments
  • The Scaling Behavior of the Front Width
    • The Saturation Width
    • Scaling Relation With the Capillary Number
    • Towards a Scaling Relation
  • Fractal Analysis of the Front
    • The Box Counting Method
    • The Density-Density Correlation Function
  • Fractal Dimension of the Mass
  • Comparison With Experimental Results
    • The Theory of Wilkinson
  • Pressure Functions and Permeability Analysis
    • The Flow Equation
    • Pressure Measurements
    • Relative Permeability
    • Towards an Understanding
• Concluding Remarks
• The Conjugate Gradient Method
• Fractals
• Scaling behavior and Power Laws
• Notation
• References
• About this document ...