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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](https://hal.science/hal-01511167)

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Submitted on 20 Apr 2017

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

K. Wojtacki<fnref n="1" />,*, L. Daridon<fnref n="1" />,<fnref n="2" />, Y. Monerie<fnref n="1" />,<fnref n="2" />

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## 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 CO₂ 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.

## Advanced Morphological Analysis of Sandstone

The starting point is **CT scan** of microstructure of **Fontainebleau sandstone** of size 256x256x256 px, where 1 px = 5.01 microns.

Numerous types of morphological descriptors: porosity, sizing (granulometry), covariance function, connectivity (tortuosity).

<figure><img src="image_4.png" /><figcaption>[Tomography - pores]</figcaption></figure>

### Granulometry Function

$$G_r(X) = 1 - \frac{|X \circ B_r|}{|X|}$$

* X – solid phase

* B_r – structuring element of size r

$$X \circ B_r = (X \ominus B_r) \oplus B_r$$

* $\ominus/\oplus$ – erosion / dilation

* $|\cdot|$ – measure

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### Summary:

$$\varphi = 0.046$$

$\varphi_{eff} = 0.029$ percolated in all directions $r_{mean} \in (17, 23)$ isotropy

## 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.

<figure><img src="image_7.png" /><figcaption>[Reconstruction method]</figcaption></figure>

**Generated samples (red box)** are validated a posteriori.

<figure><img src="image_8.png" /><figcaption>[Comparison of covariance functions]</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>[Generated microstructure]</figcaption></figure>

## Numerical Dissolution by Morphological Dilation

Chemical dissolution of porous matrix is homogeneous at sample scale [Egermann et al, 2006]. We investigate two different scenarios of dissolution:

### Isotropic

$$X_i = X_{i-1} \oplus B$$

* $X_0$ – binary image representing initial porous phase

* $X_i$ – binary image of i-th dissolution step

* $\oplus$ – morphological dilation

* B – cubic SE (3x3)

* $\Xi(\cdot)$ – operator extracting percolated network

### Percolated Network

$$X_i = [\Xi(X_{i-1}) \oplus B] + X_0$$

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /><figcaption>[Evolution of covariance functions of generated samples (red) and real microstructure (blue)]</figcaption></figure>

Constant characteristic size

Increasing characteristic size

## Elasticity - Periodic Homogenization

Method of numerical, periodic homogenisation:
**problem:** non-periodic geometry.

**solution:** additional layer of homogeneous material associated with

**fixed point method:**

$$c_{klmn}^{\text{layer}}[i] = c_{klmn}^{\text{hom}}[i - 1]$$

<figure><img src="image_12.png" /><figcaption>[External layer]</figcaption></figure>

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]:

$$P(M) = P_0 + \frac{a}{M}$$

* P(·) – computed elastic moduli

* M – resolution

* $P_0$ – searched value

<figure><img src="image_13.png" /><figcaption>[Roberts and Garboczi, 2002]</figcaption></figure>

<figure><img src="image_14.png" /></figure>

## Permeability - Elasticity Coupling

Darcy's law:

$$K = \frac{\mu Q L}{\Delta p A}$$

* K – permeability

* $\mu$ – dyn. viscosity

* Q – flux

* A – surface area

* L – sample length

* $\Delta p$ – grad. of pressure

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<footnote n="1">¹ Laboratoire de Mécanique et Génie Civil (LMGC), CNRS UMR 5508, Montpellier, France</footnote>

<footnote n="2">² Laboratoire de Micromécanique et Intégrité des Structures (MIST), UM, CNRS, IRSN, France</footnote>

<footnote>* kajetan.wojtacki@umontpellier.fr</footnote>