Chemical degradation of a numerical material - Application to a Fontainbleau sandstone

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Chemical degradation of a numerical material - Application to a Fontainbleau sandstone

Kajetan Wojtacki, Loïc Daridon, Yann Monerie

To cite this version:

Kajetan Wojtacki, Loïc Daridon, Yann Monerie. Chemical degradation of a numerical material - Application to a Fontainbleau sandstone. InterPore - 8th International Conference on Porous Media

& Annual Meeting, May 2016, Cincinnati, United States. �hal-01511167�

Elasticity - Periodic Homogenization Highlights

1. Simple and efficient method that allows to obtain morphologically
 equivalent realisations of Fontainebleau sandstone

2. New method allowing to impose periodic B.C. on non-periodic geometries

3. Two simple models of chemical degradation

4. Heuristic relation coupling elastic moduli and permeability

Context

Carbon Capture and Storage (CCS) consists of injecting large quantities of CO

2

in supercritical form directly into deeply located geological formations e.g. saline aquifers. During geological storage, chemical dissolution induces important and irreversible changes of the rock properties.

Objective is to propose a methodology which allows us to predict the evolution of effective mechanical behaviour of saline aquifers caused by microstructural changes due to CCS.




Numerical Samples - Generation

The method is inspired by natural formation process of sandstones [Bakke and Øren, 1997] and adjusted in order to

respect aforementioned morphological properties:

1. grain deposit - monodisperse grains assembly of initial radius 14 px, deposed into 3D box,

2. triaxial compaction - bulk volume reduction, 3. diagenesis - mixed uniform and random radii increment to obtain desired value of porosity.

Generated samples (red box) are validated
 a posteriori.

Method of numerical, periodic homogenisation:

problem: non-periodic geometry.

solution: additional layer of homogeneous material associated with

CT scan is naturally discretised (regular cubic mesh).

The influence of such discretization on the estimation of elastic moduli is given by [Garboczi and Day, 1995]:

M resolution

P₀ searched value [Reconstruction method]

[Comparison of covariance functions] [Generated microstructure]

P(·) elastic moduli P(·) computed

Advanced Morphological Analysis of Sandstone

The starting point is CT scan of microstructure of Fontainebleau sandstone of size 256x256x 256 px, where 1 px = 5.01 microns.

Numerous types of morphological descriptors:

porosity, sizing (granulometry), covariance function, connectivity (tortuosity).

⌅(·) operator extracting percolated network morphological dilation

X_i binary image of i-th dissolution step

B cubic SE (3x3)

X₀ binary image representing initial porous phase [Tomography - pores]

X porous phase x arbitrary point

~h translation vector

Covariance Function

P{·} probability B_r structuring element of size r

/ erosion / dilation X B_r = (X B_r) B_r X solid phase

Granulometry Function

|·| measure

' = 0.046 Summary:

'_{ef f} = 0.029 percolated in all directions r_mean 2 (17, 23) isotropy

Isotropic Percolated Network

[Evolution of covariance functions of generated samples (red) and real microstructure (blue)]

Numerical Dissolution by Morphological Dilation

Chemical dissolution of porous matrix is homogeneous at sample scale

[Egermann et al, 2006]. We investigate two different scenarii of dissolution:

[External layer]

[Roberts and Garboczi, 2002]

Increasing characteristic size Constant characteristic size

Darcy’s law:


Coupling:

K permeability µ dyn. viscosity Q flux

A surface area

p grad. of pressure L sample length

Percolated network dissolution Isotropic dissolution

Permeability - Elasticity Coupling

Percolated network dissolution Isotropic dissolution

fixed point method:

X

_i

= X

_{i 1}

B

P (K ) = 1 K

↵ K = µQL

pA P (M ) = P

₀

+ a

M

X

_i

= [⌅(X

_{i 1}

) B ] + X

₀

C (X, ~ h) = P n

x 2 X, x + ~ h 2 X o

c

^layer_klmn

[i] = c

^hom_klmn

[i 1]

G

_r

(X ) = 1 | X B

_r

|

| X |

0 0.2 0.4 0.6 0.8 1

0 10 20 30 40 50 60

Volume fraction of the solid matrix

size of opening r [pixels]

spherical SE diamond SE

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

0 10 20 30 40 50 60 70 80 90 100 110 120 130

C(X,h)

h [pixels]

Generated Samples - direction x Fontainebleau Sandstone - direction x φ² dissolution step 0 φ² dissolution step 1 φ² dissolution step 2 φ² dissolution step 3 φ² dissolution step 4

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

0 10 20 30 40 50 60 70 80 90 100 110 120 130

C(X,h)

h [pixels]

Generated Samples - direction x Fontainebleau Sandstone - direction x φ² dissolution step 0 φ² dissolution step 2 φ² dissolution step 4 φ² dissolution step 6

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4

Normalized elastic moduli

Dissolution - induced porosity GS - k

- µ FS - k - µ Arns^* - k - µ

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4

Normalized elastic moduli

Dissolution - induced porosity GS - k

- µ FS - k - µ Arns^* - k - µ

* Arns et al. 2002

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 2000 4000 6000 8000 10000 12000 14000 16000 18000

Normalized elastic moduli

Permeability K [mD]

GS - k (α = 12.13; β = 0.22) - µ (α = 9.23; β = 0.20) FS - k (α = 12.27; β = 0.22) - µ (α = 9.32; β = 0.21)

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 5000 10000 15000 20000 25000 30000 35000 40000

Normalized elastic moduli

Permeability K [mD]

GS - k (α = 15.12; β = 0.22) - µ (α = 10.87; β = 0.20) FS - k (α = 16.57; β = 0.23) - µ (α = 11.85; β = 0.21) 0

0.01 0.02 0.03 0.04 0.05

0 10 20 30 40 50 60 70 80 90 100 110 120 130

Cov(X,h)

h [pixels]

GS - x FS - x φ²

0 0.002 0.004 0.006 0.008 0.01

0 10 20 30 40

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

0 15 30 45 60 75 90 105 120

Cov(X,h)

h [pixels]

X - direction Y - direction Z - direction φ²

0 0.002 0.004 0.006 0.008 0.01

0 15 30

K. Wojtacki

^1,*

, L. Daridon

^1,2

, Y. Monerie

^1,2

1

Laboratoire de Mécanique et Génie Civil (LMGC), CNRS UMR 5508, Montpellier, France

2

Laboratoire de Micromécanique et Intégrité des Structures (MIST), UM, CNRS, IRSN, France

CHEMICAL DEGRADATION OF A NUMERICALLY GENERATED MATERIAL - APPLICATION FOR FONTAINEBLEAU SANDSTONE

* kajetan.wojtacki@umontpellier.fr