1. Domain Preparation
   1. Synthetic Media Generator, or
   2. Imaging of Real Porous Media (micro-ct, FIB-SEM, etc)
2. Modeling & Simulations
3. Experiments to Calibrate

A virtual porous media can be created from two different methods:

1. Actual Imaging of the Porous Media (core) with technology such as Miro-CT or FIB-SEM (for tight core).
2. Pore Size Distribution Data, which is used to create a virtual packing of particles.

Micro-CT Images

Sphere Swelling Method for Synthetically Derived Porous Media

Generated Synthetic Media Examples

Validation of Results from Virtual Packing of Spheres

Simulator was modified to reduce the porosity to lower values by putting new grains to empty spaces of original generated pattern.

• Different layer each one has its own Pore Size Distribution (PSD) and porosity
• Layering could be horizontal or vertical
• Each layer could have its own specific thickness

First principle calculations (e.g, Navier-Stokes, Maxwell, etc) are used to derive properties such as:

• Permeability
• Porosity
• NMR spectra
• Thermal Conductivity
• Electrical Resistivity

Sand pack from actual set of images.  However in this case we synthetically created more grains in each 2D cross-section in order to decrease its porosity and evaluate Archie exponent shown in the below.

• From Bloch-Torrey equations:  (∂c_i)/∂t + ∇.(-D∇c_i) = – c_i/T_2B ,
• With boundary conditions:       ρ_s c_i + D_i  n ̂ ∇c_i = 0
  • D_i: Diffusion coefficient (constant in this, however generally is tensor)
  • c_i: concentration
  • ρ_s: surface relaxivity and is a measure of loss of magnetization at fluid boundary S.
  • T_2B is Bulk relaxation time:

For water T_2B=3.1 s, water diffusion coefficient: 2.07 x 〖10〗^(-9) m^2/s .

For Oil T_2B= 0.00122 s, Oil viscosity: 1000 cp,

Oil Diffusion coefficient: 1.3 x 〖10〗^(-12) m^2/s.

〖water-solid:  ρ〗_s = 4.1 x 〖10〗^(-5) m/s,  〖oil-solid:  ρ〗_s = 4.1 x 〖10〗^(-6) m/s

Water – oil interface:  ρ_s= 4.1 x 〖10〗^(-7) m/s

• FOR LIQUID SOLVENTS CT IS MORE THAN ADEQUATE
• FOR GASEOUS SOLVENT MRI IS MORE APPROPRIATE
• BUT FIRST IT NEEDS TO BE TESTED WITH LIQUIDS

• 1-D Nuclear Magnetic Resonance Imaging (MRI) is employed to obtain diffusivity data for a toluene-heavy oil system
• “Oxford Instruments” NMR apparatus operating at a resonance frequency of 2.5MHz is used

• The diffusion experiment was monitored for 20 days
• The width of the signal: the length of the sample
• The height of the signal: the amplitude in each cross section

• Diffusion coefficient was obtained from concentration profiles and solving the diffusion equation
• Diffusion coefficient is a strong function of concentration
• PLEASE NOTE THE DIFFERENCE IN SHAPE
• THIS IS A DIRECT RESULT OF THE INTERPRETATION MODEL USED
• Different MODELS ARE TESTED

Solving equations of flow and transport:

• Continuity equatio

∇.(ρu)=0

• Navier-Stokes equation

ρ ∂u/∂t+ρ(u.∇)u=∇.[-pI+μ∇u]+ρg

• Convection-Diffusion equation

∂c/∂t+∇.(-D∇c)+u.∇c=0

Simulations are being conducted at various flow conditions:

• Wide range of displacement velocity in terms of Peclet number (N_P)
• Various viscosity ratios in terms of log-viscosity ratio (R=ln⁡(μ_displaced/μ_displacing ))
• Incorporation of gravity effects
• Incorporation of concentration dependent diffusion coefficient
• Multicomponent mass transfer

Mixing zone growth rate (σ) at different flow conditions:

• Effect of Peclet number and viscosity ratio

• Dispersion coefficient calculation:

At macro-scale, longitudinal dispersion can be described by a convection-dispersion equation:

∂c/∂t+∇.(-K_L ∇c)+u.∇c=0

K_L in a porous media along the flow direction can be calculated by matching the numerically obtained concentration profile at the outlet with the analytical semi-infinite solution of convection-dispersion equation.

• Effect of Viscosity ratio

• Concentration shock or continual interface renewal at the interface between the undiluted heavy oil and solvent vapour can be one of the reasons for higher production rates than expected
• Low-viscosity diluted heavy oil adjacent to the interface flows away from the interface faster than the fluid diffuses into the heavy oil
• Three forces acting on a drop/slug/film of fluid within a porous medium: Gravity, Interfacial Tension and Viscous Forces
• Gravity must overcome interfacial forces during gravity drainage to cause the fluid to move. Only once the fluid is moving, viscosity come into play

SAGD Modelling at the pore level

Numerous experiments

What is the thickness of condensate?

Phase changes in a steam flood

T_sat=100 ℃

Drainage and Imbibition of viscous oil and water.

T = 140°C and v = 1 x 10-3 m/s

What can we do when images are not available?

How about when we want quick and dirty results?

How about when we have limited computer capabilities?

Pore Network Modelling IS FAST BUT TOO SIMPLE

CAN WE APPLY THE RULES OF PNM INTO DIRECT PORE SPACE RECONSTRUCTIONS?

Generate all two-phase properties using the hierarchical rules but the full morphology

Currently working on combining CFD and PM for complex geometries

Extract images of core and center them.

Mask different rock types.

Apply properties, physics and solve in CFD Software

Assumptions and Input Parameters

Results