# Digital Core Analysis

# Workflow

- Domain Preparation
- Synthetic Media Generator, or
- Imaging of Real Porous Media (micro-ct, FIB-SEM, etc)

- Modeling & Simulations
- Experiments to Calibrate

# Generate Porous Media

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

- Actual Imaging of the Porous Media (core) with technology such as Miro-CT or FIB-SEM (for tight core).
- 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

# Reservoir Characterization Using NMR

# 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.

# Diffusion Coefficient Measurement Using Computed Tomography and Nuclear Magnetic Resonance Imaging

# Evaluation of Diffusion of Light Hydrocarbon Solvents in Bitumen

# 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

# Miscible Displacements

# Modelling Process

Solving equations of flow and transport:

- Continuity equatio

∇.(ρ**u**)=0

- Navier-Stokes equation

ρ ∂**u****/**∂t+ρ(**u**.∇)**u=**∇.[-p**I**+μ∇**u****]**+ρ**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

# Effect of IFT

# The Case of 0.015mN/m

# Phase Change Phenomena

SAGD Modelling at the pore level

Numerous experiments

What is the thickness of condensate?

Phase changes in a steam flood

# Mineral Heterogeneity

# 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

# Geometry

# Fluid Properties

Table 5-2: Fluid Properties

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

(Svrcek and Mehrotra, 1982)

# Boundary and Initial Conditions

# Case Studies

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.

# Drainage Demonstration – Viscosity Ratio 1:1

# 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

# Berea Sandstone Network Reconstruction

# Network Calibration & Validation

# Capillary Pressure Curves

# 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

# Flow Properties

# 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

# Experimental Validation

# Two-Phase Simulations

**Mixed-wet**

**Fractional-Wet**

**Precipitation**

# Three -Phase Simulations

# Effective Properties

Thermal Conductivity Calculations

Electrical Resistivity Calculations

# Measurement Technique and Procedure for Thermal Conductivity

# 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.

# From Micro Scale to Macro Scale

# CT Images

Extract images of core and center them.

Raw CT Image

Processed CT Image

# 3-D Reconstruction of CT

Mask different rock types.

Processed CT Image

Masked CT Image

# 3-D Reconstruction of Core

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