# Workflow

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

# Generate Porous Media

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

# Tight Media

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

# Multi-layered Media

• 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

# Single Phase Properties

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

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

# Single Phase Flow Properties for Sandstone

13% porosity

Resistivity is 80 Ω m

Permeability is 40 µD

# F, Archie’s Exponents

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.

φ ≅ 50%

φ ≅ 18%

φ ≅ 42%

φ ≅ 8%

F = A “φ” ^(-m)   for different φ,  gives m≅1.7

# NMR Relaxation in Porous Media

• 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

Porous media saturated if water (blue) and oil (black)

NMR spectra Fully saturated with water (blue) Fully saturated with oil (black)

Oil-Water system resulting from addition of the different components.

# Apparatus for MRI

• 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

# Acquiring MRI signals

• 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

# Converting MRI Signals to Concentration Profiles

MRI Signals

Calibration Curve

Oil concentration profiles at different mixing times

# Diffusion Coefficients

• 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
• 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

# Modelling Process

Solving equations of flow and transport:

• Continuity equatio

∇.(ρu)=0

• Navier-Stokes equation

ρ ∂u/∂t+ρ(u.∇)u=∇.[-pI+μ∇u]g

• Convection-Diffusion equation

∂c/∂t+∇.(-D∇c)+u.∇c=0

# Flow Properties

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

# Modelling Outcomes

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

Mixing zone length calculation:

• 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

# Partial Miscibility

• 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
• Solve transport of species in continuous liquid and gas phases and across fluid interfaces (convection-diffusion equation)
• Based on color function volume-of-fluid (CF-VOF) method
• Each computational cell stores pressure, velocity, phase volume fraction and concentration of species (solvent)
• Viscosity of the heavy oil and solvent mixture was modeled as a function of concentration of diffused solvent in heavy oil

# Phase Change Phenomena

SAGD Modelling at the pore level

Numerous experiments

What is the thickness of condensate?

Phase changes in a steam flood

# Experiment vs. Modelling; Lab SAGD

Modelling

Experiment

Theory

• Viscosity ratio: 10000 at 25oC
• Density ratio: 1
• Neutral wet
• Saturation Temperature: 373.15K
• Steam Temperature: 403.15K
• Oil Temperature: 298.15K
• IFT: Steam-Water=0.071 Nm-1
• IFT: Oil-Water=0.0341 Nm-1
• IFT: Steam-Oil=0.024 Nm-1

# Fluid Properties

Table 5-2: Fluid Properties

Figure 5-3: (a) Viscosity functionality, (b) Density functionality to temperature

(Svrcek and Mehrotra, 1982)

T_sat=100 ℃

# Saturation Profiles

Case #1

• Highly efficient heat transfer reduces the accumulation of condensate

Case #3

# Dynamic Simulations

Drainage and Imbibition of viscous oil and water.

# Demonstration of Imbibition

T = 140°C and v = 1 x 10-3 m/s

• High shear flow
• Local Ca numbers are high enough to result in blob deformation and breakage
• Viscosity Ratio 196

# Oil Ganglia Dynamics

• Choke-off is the main mechanism of ganglia break up observed in the runs (120 – 122 ms)
• Oil mobilization happens through successive jumps (122 – 124 ms)
• Once it breaks up, smaller ganglia can get mobilized or stranded, depending on external pressure gradient and local capillary pressure

# Wettability Heterogeneity

• Fractional-wet
• Dynamic Solver
• Coupled Level Set Volume of Fluid (CLSVOF)
• OpenFoam CFD package

# 3D VOF-based Simulations

• Unconsolidated sand packing of 1600 grains
• Creation of virtual porous medium with a given particle size distribution
• Numerical simulation of immiscible displacement of primary drainage at different viscosity ratios: In this case the viscosity ratio R=10

# Pore-Network Modelling

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 Space Simplification

Random Network Model

• Pore bodies interconnected by throats
• Pore Bodies are randomly positioned throughout the space.

Pore Bodies:

• Size distribution
• Shape factor distribution
• Connectivity number distribution

Pore Throats:

• Size distribution
• Shape factor distribution
• Length distribution

# Algorithm

PROCESS:

P_c= P_(non-wetting phase)-P_(wetting phase)

Pressure of the non-wetting phase can increase leading to:

• Increase of Pc during drainage

Pressure of the wetting phase can increase leading to:

• Decrease of Pc during imbibition

# Green River Basin

• Pore throat radius distribution is extracted from mercury porosimetry data.
• Pore throat length distribution is obtained by iteration to match formation factor
• Pore body distribution is obtained by iteration to match porosity and permeability
• Connectivity number distribution is obtained by iteration to match experimental capillary pressure curve

# Pore Morphological Simulations

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

Mixed-wet

Fractional-Wet

Precipitation

# Effective Properties

Thermal Conductivity Calculations

Electrical Resistivity Calculations

# Multi-Scale Simulations

Currently working on combining CFD and PM for complex geometries

# Simulation Packages / Software

CFD solvers were implemented in order to:

• Simulate multi-phase isothermal scenarios.
• Simulate thermal processes applying phase change models.
• Simulate mass transfer along the interface during solvent injection.

PNM simulator was adapted in order to:

• Simulate multi-phase capillary dominant displacements.

PM toolbox was developed in order to:

• Simulate multi-phase capillary dominant displacements.
• Predict effective petrophysical, thermal and mechanical properties.
• Simulate multi-sale displacements in a time efficient manner.

# CT Images

Extract images of core and center them.

Raw CT Image

Processed CT Image

# 3-D Reconstruction of CT

Processed CT Image

Shaly Sand

Full Geometry

# Multiphysics Modelling of Core and Rock

Apply properties, physics and solve in CFD Software

Temperature Profile

Pressure Profile

# Real Core Example

• Full core decomposed into sub cores
• For each sub core thermal conductivity and permeability was computed
• Using upscaling algorithm properties of full length core were calculated

## Assumptions and Input Parameters

• Two domains: fluid and rock
• Two rock types: sand and shaly sand
• Geometric mean mixing rule for effective thermal conductivity
• Constant permeability for each rock type