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Rock Mechanics and Rock Engineering

, Volume 52, Issue 12, pp 5099–5121 | Cite as

Multiscale Poromechanical Modeling of Shales Incorporating Microcracks

  • Vasav Dubey
  • Mohammad Mashhadian
  • Sara AbediEmail author
  • Arash NoshadravanEmail author
Original Paper
  • 239 Downloads

Abstract

A probabilistic multiscale model for poroelastic properties of organic-rich shale is extended to include the effect of microcracks. The approach is based on physics-based modeling of poroelastic properties and uncertainty propagation from nano- to macro-scale using homogenization schemes. The multiscale thought model used in this work improves on an existing thought model by adding another level to model embedded microcracks. From the scale of single clay particle to the scale of embedded microcracks, the model parameters are calibrated using experimental characterization at different length scales. To quantify the crack density and microcrack orientations, an inverse optimization approach is utilized which uses the UPV measurements at the macro-scale. Major model parameters representing compositional and mechanical properties at each length scale are modeled as a random variable. Probabilistic description of both scalar random variables and matrix-valued random variables is constructed using the maximum entropy principle (MaxEnt). Using such statistical description of model input parameters with Monte Carlo simulation, probabilistic descriptions of model outputs are obtained at each length scale. The results of stochastic upscaling are validated against available experimental values. Finally, a global sensitivity analysis is performed to identify the model input parameters which are most influential to the statistical fluctuation of upscaled poroelastic properties. The presented multiscale framework provides an enhanced understating regarding the effect of uncertainties associated with microcrack density and orientation, and other subscale features on homogenized poroelastic properties, thus further improves predictive capability for shale multiscale behavior.

Keywords

Micro-poromechanics Organic-rich shale Probabilistic modeling Microcrack-induced damage 

List of symbols

\(\left[ \varSigma \right]\)

Stress matrix

\(\left[ E \right]\)

Strain matrix

\(\left[\kern-0.15em\left[ C \right]\kern-0.15em\right]\)

Fourth-order stiffness tensor

\(\left[ \alpha \right]\)

Second-order tensor of Biot pore pressure coefficients

\(p\)

Pore pressure

ϕ

Porosity

\(N\)

Solid Biot modulus

\(\left[\kern-0.15em\left[ A \right]\kern-0.15em\right]\)

Strain localization tensor

\(\varOmega\)

Representative volume element

\(\left[\kern-0.15em\left[ I \right]\kern-0.15em\right]\)

Fourth-order identity tensor

\(\left[\kern-0.15em\left[ P \right]\kern-0.15em\right]\)

Fourth-order Hill polarization tensor

\(M\)

Overall Biot modulus

\(\eta^{i} , f^{i}\)

Volume fraction of the ith phase at Level I and Level II, respectively

\({\mathcal{N}}\)

Number of cracks per unit volume

\(a\)

Crack radius

\(c\)

Crack width

\(\varepsilon\)

Crack aspect ratio

\(d\)

Crack density parameter

\(M_{1} ,M_{3}\)

Indentation moduli parallel to the bedding plane and normal to the bedding plane, respectively

\(C_{ij}\)

Component of stiffness matrix in Voigt’s notation

\(\left\| \cdot \right\|_{\text{F}}\)

Frobenius norm

\(K\)

Bulk modulus

\(\nu\)

Poisson’s ratio

\(G\)

Shear modulus

\(\Delta_{\text{itz}}\)

Thickness of interfacial transition zone (ITZ)

\(C^{\text{itz}}\)

Coefficient for elastic properties of ITZ

\({\text{NRMSD}}\)

Normalized root mean square deviation

\(\varepsilon , \gamma , \delta^{*}\)

Thomsen’s anisotropy parameters

\(E_{x} , E_{z}\)

Young’s modulus parallel to the bedding plane and normal to the bedding plane, respectively

\(\lambda_{0} , \lambda_{1} , \lambda_{2} ,\lambda^{\text{sol}} ,\lambda_{1}^{\text{sol}} , \ldots ,\lambda_{5}^{\text{sol}}\)

Lagrange multipliers

Notes

Acknowledgements

The authors acknowledge partial funding from the Crisman-Berg Hughes Institute at the Department of Petroleum Engineering of Texas A&M University. Computational resources provided by Texas A&M High Performance Research Computing were utilized to conduct portions of this research. Furthermore, the authors thank the reviewer and the guest editor for providing insightful comments and feedback which improved the quality of the work.

Funding

This study was partially funded by the Crisman-Berg Hughes Institute at the Department of Petroleum Engineering of Texas A&M University (Project no. 2.07.16).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Zachry Department of Civil EngineeringTexas A&M UniversityCollege StationUSA
  2. 2.Ponce-Fuess Engineering, LLCDallasUSA
  3. 3.Harold Vance Department of Petroleum EngineeringTexas A&M UniversityCollege StationUSA

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