#
Advanced Geomechanical Model to Predict the Impact of CO_{2}-Induced Microstructural Alterations on the Cohesive-Frictional Behavior of Mt. Simon Sandstone

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_{2}Sequestration: Mechanisms and Coupling Behaviours)

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Materials

#### 2.2. CO${}_{2}$ Alteration Experiments

#### 2.3. Grinding and Polishing, Optical Microscopy, and Scanning Electron Microscopy Imaging

#### 2.4. Grid Nanoindentation Testing

## 3. Poromechanical Model

#### 3.1. Conceptual Multiscale Model for Host Rock

#### 3.2. Upscaling of Poro-Elastic Properties

#### 3.2.1. Molecular Level (0) → Pore Level (I)

#### 3.2.2. Pore Level (I) → Core Level (II)

#### 3.3. Casta neda and Willis Variational Framework for Strength Upscaling

#### 3.3.1. Molecular Level (0) → Pore Level (I)

#### 3.3.2. Pore Level (I) → Core Level (II)

#### 3.3.3. Hardness-Packing Density Relations at Pore Level (I)

## 4. Results

#### 4.1. CO${}_{2}$-Induced Micro-Structural Changes at Pore Level (I) from SEM Imaging

#### 4.2. Calibration and Validation of the Theoretical Model

#### 4.3. CO${}_{2}$-Induced Micro-Structural Changes at Pore Level (I) Using Nanoindentation Testing Integrated with Micromechanical Modeling

#### 4.4. Changes in Cohesive-Frictional Response at the Core Level (II)

## 5. Discussion

#### 5.1. Origins of Mechanical Strength in Mt. Simon Sandstone

#### 5.2. CO${}_{2}$-Induced Geochemical Reactions

#### 5.3. Implications for CO${}_{2}$-Induced Microseismicity

#### 5.4. Induced Microseismicity Within the Illinois-Basin Decatur Project

## 6. Conclusions

- Incubation with CO${}_{2}$-saturated brine, under geological conditions, results in a significant increase in the microporosity and macroporosity.
- CO${}_{2}$-induced microstructural changes result in a sharp decrease in the size of the yield surface, and in the value of the friction coefficient.
- The reduction in the friction coefficient, following CO${}_{2}$-induced geochemical alterations, leads to a drop in the threshold pore-fluid pressure and an increase in the risk of fault slip.
- Using the Illinois Basin Decatur Project as a case study, our investigation showed that CO${}_{2}$-induced geochemical reactions, causing changes in nano- and macroporosity, are a plausible explanation for the surge in microseisimic activity observed during CO${}_{2}$ injection in the Mt. Simon formation and for the observed residual microseismicity after cessation of the CO${}_{2}$ injection activities.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

${\alpha}_{s}$ | internal friction angle of solid skeleton |

${A}^{i}$ | long axis of failure envelope at level i |

$\langle {\mathbb{A}\rangle}_{j}^{i}$ | strain concentration tensor of phase j at level i |

${B}^{i}$ | short axis of failure envelope at level i |

${\mathit{b}}^{I}$ | Biot coefficient tensor at Level I |

${\mathit{b}}_{I/II}^{II}$ | Biot coefficient tensors at Level II |

${c}_{s}$ | cohesion of solid skeleton |

${\mathbb{C}}_{s}$ | stiffness tensor of skeleton at Level 0 |

${\mathbb{C}}^{i}$ | effective stiffness tensor at level i |

${D}_{v}$ | volumetric strain rate |

${D}_{d}$ | deviatoric strain rate |

$\mathbf{\delta}$ | grain aspect ratio |

$\mathit{E}$ | global strain tensor |

${f}^{i}$ | yield function at level i |

$\eta $ | packing density of solid phase at Level I |

H | indentation hardness |

$\mathbb{I}$ | fourth-order identity tensor |

$\mathbb{J}$ | spherical projection fourth order tensor |

${\kappa}_{s}$ | bulk modulus of solid skeleton at Level 0 |

${\kappa}^{i}$ | bulk modulus at level i |

${\kappa}^{i}$ | linear upscaling function for the bulk modulus at level i |

$\mathbb{K}$ | deviatoric projection fourth order tensor |

${m}_{s}$ | plane strain indentation modulus of solid skeleton |

${\mu}_{s}$ | shear modulus of solid skeleton at Level 0 |

${\mu}^{i}$ | shear modulus at level i |

${\mathcal{M}}^{i}$ | linear upscaling function for the shear modulus at level i |

M | indentation modulus |

${\nu}_{s}$ | Poisson’s ratio of solid skeleton |

${N}^{I}$ | skeleton Biot modulus at Level I |

${N}_{ij}^{II}\phantom{\rule{0.166667em}{0ex}}i,j\in \{1,2\}$ | skeleton Biot moduli at Level II |

${\mathcal{P}}_{\mathrm{slip}}$ | risk of slip on a cohesionless fault |

${P}_{f}$ | pore fluid pressure |

${P}_{{f}_{\mathrm{thr}}}$ | threshold pore fluid pressure |

${\mathsf{\Pi}}^{i}$ | plastic dissipation potential at level i |

$\varphi $ | microporosity |

$\phi $ | nanoporosity |

${S}^{i}$ | eccentricity of failure envelope at level i |

$\mathsf{\Sigma}$ | global stress tensor |

${\mathsf{\Sigma}}_{m}$ | global volumetric stress |

${\mathsf{\Sigma}}_{d}$ | global deviatoric stress |

$\tau $ | prestress |

${\nu}_{s}$ | non-linearity function of solid skeleton at Level 0 |

${W}_{0}^{i}$ | homogenized strain rate energy at level i |

## Appendix A. Statistical Deconvolution Analysis of Indentation Data

**Figure A1.**Phase distribution after analyzing the grid indentation tests performed on unaltered and altered Mt. Simon sandstone specimens.

**Figure A2.**Spatial phase distribution after analyzing the grid indentation tests performed on unaltered and altered Mt. Simon sandstone specimens.

**Table A1.**Phase distribution following grid indentation of unaltered and altered Mt. Simon sandstone specimens.

Material | Apparent Microporosity (%) | Apparent Feldspar Content (%) | Quartz Content (%) | Siderite Content (%) |
---|---|---|---|---|

6925-U | 16.6 | 30.5 | 52.9 | 0 |

6925-AS1 | 25.3 | 20.4 | 45.1 | 9.2 |

6927.55-U | 16.7 | 21.8 | 57.0 | 4.5 |

6927.55-AS1 | 20.7 | 39.8 | 39.5 | 0 |

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**Figure 1.**(

**a**) Location of the verification well 1 in the Illinois Basin - Decatur Project. Adapted from [2]. (

**b**) Digital photography image of an unaltered Mt. Simon sandstone specimen. Credits: Ange-Therese Akono, Northwestern University, 2018.

**Figure 3.**Optical microscopy images of unaltered Mt. Simon sandstone: (

**a**) 6925-U sample; and (

**b**) 6927.55-U sample.

**Figure 4.**Microstructural characterization of unaltered Mt Simon Sandstone samples, 6925-U: (1) quartz grain (dark gray); (2) K-feldspar grain (light gray); (3) clay; (4) macropore; and (5) micropore. Magnification: (

**a**) 55×; (

**b**) 1054×; (

**c**) 1149×; and (

**d**) 6361×.

**Figure 6.**Conceptual multiscale model of Mt Simon Sandstone. Level 0 is the molecular level. Level I is the Pore level. Level II is the Core level. Level III is the Formation level.

**Figure 7.**Hardness-packing density relationships at Level I. (

**a**) Numerical simulations of indentation test: initial configuration. (

**b**) Close-up of the deformed configuration for indentation test simulation. The yellow triangle represents the rigid conical indenter. (

**c**) Theoretical indentation modulus-packing density curve for ${\nu}_{s}=0.25$. (

**d**) Theoretical indentation hardness—packing density curves for ${\nu}_{s}=0.25$ and various values of ${\alpha}_{s}$. H is the indentation hardness, M is the indentation modulus, ${m}_{s}$ is the plane strain modulus for the solid skeleton, ${c}_{s}$ is the solid skeleton cohesion, ${\nu}_{s}$ is the solid skeleton Poisson’s ratio, and ${\alpha}_{s}$ is the solid skeleton coefficient of internal friction.

**Figure 8.**(

**a**) Backscattered electron microscopy image of Mt. Simon sandstone sample 6925-U; and (

**b**–

**d**) backscattered electron microscopy images of Mt. Simon sandstone sample 6925-AS1. The red arrows indicate channel formation whereas the dotted circles underscore an increase in pore throat size. Quartz grains are shown in dark gray and feldspar grains are shown in light gray. Magnification: (

**a**) 55×; (

**b**) 78×; and (

**c**) 283×.

**Figure 9.**Experimental calibration of theoretical model at Level I with nanoindentation tests on Mt. Simon sandstone 6925-U ($N=441$ indentation tests). (

**a**) M is the indentation modulus, (

**b**) H is the indentation hardness. $\eta =1-\phi $ is the local solid packing density, and $\varphi $ is the local microporosity. The blue dots represent the experimental data points, whereas the solid line displays the theoretical solution. ${R}^{2}$ is the coefficient of determination and $RMSE$ is the root mean square error.

**Figure 10.**Experimental validation of theoretical model at Level I with nano-indentation tests on Mt. Simon sandstone 6925-AS1 ($N=441$ indentation tests). (

**a**) M is the indentation modulus, (

**b**) H is the indentation hardness. $\eta =1-\phi $ is the local solid packing density, and $\varphi $ is the local microporosity. The blue dots represent the experimental data points whereas the solid line displays the theoretical solution. ${R}^{2}$ is the coefficient of determination and $RMSE$ is the root mean square error.

**Figure 11.**Experimental validation of theoretical model at Level I with nanoindentation tests on Mt. Simon sandstone 6927.55-U ($N=441$ indentation tests). (

**a**) M is the indentation modulus, (

**b**) H is the indentation hardness. $\eta =1-\phi $ is the local solid packing density, and $\varphi $ is the local microporosity. The blue dots represent the experimental data points, whereas the solid line displays the theoretical solution. ${R}^{2}$ is the coefficient of determination and $RMSE$ is the root mean square error.

**Figure 12.**Experimental validation of theoretical model at Level I with nanoindentation tests on Mt. Simon sandstone 6927.55-AS1 ($N=441$ indentation tests). (

**a**) M is the indentation modulus, (

**b**) H is the indentation hardness, $\eta =1-\phi $ is the local solid packing density, and $\varphi $ is the local microporosity. The blue dots represent the experimental data points whereas the solid line displays the theoretical solution. ${R}^{2}$ is the coefficient of determination and $RMSE$ is the root mean square error.

**Figure 13.**Measured probability distribution of the local packing density $\eta $ in: (

**a**) unaltered Mt. Simon sandstone 6925-U; and (

**b**) altered Mt. Simon sandstone 6925-AS1. The vertical line marks the local packing density threshold between macropores and micropores.

**Figure 14.**Measured probability distribution of the local packing density $\eta $ in: (

**a**) unaltered Mt. Simon sandstone 6927.55-U; and (

**b**) altered Mt. Simon sandstone 6927.55-AS1. The vertical line marks the local packing density threshold between macropores and micropores.

**Figure 15.**Predicted alteration in macroscopic failure envelope in drained conditions for (

**a**) Mt. Simon sandstone at 6925 ft. and (

**b**) 6927.55 ft. following incubation into CO${}_{2}$-saturated brine. ${\mathsf{\Sigma}}_{m}$ is the global volumetric stress whereas ${\mathsf{\Sigma}}_{d}$ is the global deviatoric stress.

**Figure 16.**Changes in threshold pore fluid pressure ${P}_{{f}_{thr}}$ due to exposure to CO${}_{2}$-saturated brine: (

**a**) Mt. Simon sandstone 6925 ft.; and (

**b**) Mt. Simon sandstone 6927.55 ft. The solid green line indicates the magnitude of the pore fluid pressure before CO${}_{2}$ injection, as recorded in Zone 3 in October 2011. The dotted black line indicates the maximum level of the pore fluid pressure during injection, that maximum was reached in January 2013 [2,44]. The associated case study is the Illinois Basin Decatur Project.

**Figure 17.**Case study of Zone 3 (6946 ft.) of the Illinois Basin Decatur Project: risk of slip ${\mathcal{P}}_{\mathrm{slip}}$ of hypothetical faults located at depths 6925 ft. and 6927.55 ft. (

**a**) Pre- CO${}_{2}$-injection conditions (October 2011): ${P}_{{f}_{0}}$ = 3165 psi (21.82 MPa). (

**b**) CO${}_{2}$ Peak injection conditions (February 2012): ${P}_{f}$ = 3340 psi (23.03 MPa) and $\Delta {P}_{f}=5.5\%\phantom{\rule{0.166667em}{0ex}}{P}_{{f}_{0}}$. (

**c**) CO${}_{2}$ injection conditions (January 2013): ${P}_{f}=3325$ psi (22.93 MPa) and $\Delta {P}_{f}=5\%\phantom{\rule{0.166667em}{0ex}}{P}_{{f}_{0}}$. (

**d**) Post-CO${}_{2}$ injection conditions (June 2015 and seven months after the end of CO${}_{2}$ injection operations): ${P}_{f}=3207$ psi (22.11 MPa) and $\Delta {P}_{f}=1.3\%{P}_{{f}_{0}}$. The red dots indicate orientations for potential fault slips.

Sample Notation | Burial Depth | Alteration Procedure |
---|---|---|

6925-U | 6925 ft. | None |

6925-AS1 | 6925 ft. | Incubation in CO${}_{2}$-saturated brine at 50 ${}^{\circ}$C and 17.2 MPa for one week |

6927.55-U | 6927.55 ft. | None |

6927.55-AS1 | 6927.55 ft. | Incubation in CO${}_{2}$-saturated brine at 50 ${}^{\circ}$C and 17.2 MPa for one week |

Level | Mechanical Constants | Structural Parameters |
---|---|---|

0 | $({m}_{s},{\nu}_{s},{c}_{s},{\alpha}_{s})$ | |

I | $({\mathbb{C}}^{I},{\mathit{b}}^{I},{N}^{I},{f}^{I})$ | $\phi $ |

II | $({\mathbb{C}}^{II},{\mathit{b}}_{I}^{II},{\mathit{b}}_{II}^{II},{N}_{i,j\in \{1,2\}}^{II},{f}^{II})$ | ($\varphi $,${\varphi}_{i}$, $\delta $) |

**Table 3.**Measured alterations in pore structure in Mt. Simon sandstone following incubation in CO${}_{2}$-saturated brine.

Material | Macroporosity $\mathit{\varphi}$ | Microporosity $\mathit{\phi}$ |
---|---|---|

6925-U | 0.181 | 0.083 |

6925-AS1 | 0.224 | 0.096 |

6927.55-U | 0.147 | 0.105 |

6927.55-AS1 | 0.245 | 0.111 |

**Table 4.**Model parameters. For each sample—Mt. Simon sandstone 6925 ft. altered and unaltered and Mt. Simon sandstone 6927.55 altered and unaltered—the phase distribution $\left({\varphi}_{i}\right)$ is given by application of Gaussian mixture modeling to the grid indentation data; meanwhile, the macroporosity $\varphi $ and the microporosity $\phi $ is given by application of our nonlinear micromechanics model. The values of the macroporosity and microporosity are displayed in Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13 and summarized in Table 3.

Level | Model Parameters |
---|---|

0 | $({m}_{s}=70.95\phantom{\rule{0.166667em}{0ex}}\mathrm{GPa},{\nu}_{s}=0.25,{c}_{s}=0.37\phantom{\rule{0.166667em}{0ex}}\mathrm{GPa},{\alpha}_{s}=0.12)$ |

I | $\phi $ |

II | $(\varphi ,{\varphi}_{i},\delta =0.022)$ |

**Table 5.**Predicted alterations in drained macroscopic elasto-plastic constants in Mt. Simon sandstone following incubation in CO${}_{2}$-saturated brine. ${S}^{II}$ is the offset. ${\alpha}^{II}={B}^{II}/{A}^{II}$ is the macroscopic drained friction coefficient and ${\theta}^{II}=\mathtt{arctan}\left({\alpha}^{II}\sqrt{3}\right)$ is the friction angle. ${A}^{II}$ (respectively, ${B}^{II}$) is the long axis (respectively, short axis) of the macroscopic yield curve for drained conditions.

Material | ${\mathit{E}}^{\mathit{II}}$ (GPa) | ${\mathit{S}}^{\mathit{II}}$ (MPa) | ${\mathit{\alpha}}^{\mathit{II}}$ | ${\mathit{\theta}}^{\mathit{II}}$ (${}^{\circ}$) | ${\mathit{B}}^{\mathit{II}}$ (MPa) |
---|---|---|---|---|---|

6925-U | 23.89 | −3.45 | 0.50 | 40.76 | 338.57 |

6925-AS1 | 20.32 | −4.37 | 0.45 | 37.71 | 297.91 |

6927.55-U | 30.34 | −4.04 | 0.54 | 42.95 | 379.84 |

6927.55-AS1 | 15.93 | −10.18 | 0.39 | 33.79 | 229.10 |

**Table 6.**Predicted threshold pore fluid pressure ${P}_{{f}_{\mathrm{thr}}}$ and critical cohesionless fault orientation for Mt. Simon sandstone in both unaltered and altered states.

Material | ${\mathit{P}}_{{\mathit{f}}_{\mathbf{thr}}}$ (MPa) | ${\mathit{P}}_{{\mathit{f}}_{\mathbf{thr}}}$ (psi) | Critical Fault Orientation (${}^{\circ}$) |
---|---|---|---|

6925-U | 26.31 | 3815 | ±33 |

6925-AS1 | 23.65 | 3430 | ± 34 |

6927.55-U | 27.87 | 4042 | ± 33 |

6927.55-AS1 | 19.39 | 2812 | ± 35 |

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Akono, A.-T.; Werth, C.; Shi, Z.; Jessen, K.; Tsotsis, T.T.
Advanced Geomechanical Model to Predict the Impact of CO_{2}-Induced Microstructural Alterations on the Cohesive-Frictional Behavior of Mt. Simon Sandstone. *Minerals* **2021**, *11*, 38.
https://doi.org/10.3390/min11010038

**AMA Style**

Akono A-T, Werth C, Shi Z, Jessen K, Tsotsis TT.
Advanced Geomechanical Model to Predict the Impact of CO_{2}-Induced Microstructural Alterations on the Cohesive-Frictional Behavior of Mt. Simon Sandstone. *Minerals*. 2021; 11(1):38.
https://doi.org/10.3390/min11010038

**Chicago/Turabian Style**

Akono, Ange-Therese, Charles Werth, Zhuofan Shi, Kristian Jessen, and Theodore T. Tsotsis.
2021. "Advanced Geomechanical Model to Predict the Impact of CO_{2}-Induced Microstructural Alterations on the Cohesive-Frictional Behavior of Mt. Simon Sandstone" *Minerals* 11, no. 1: 38.
https://doi.org/10.3390/min11010038