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Two-Scale Geomechanics of Carbonates

  • Wenfeng Li
  • A. Sakhaee-Pour
Original Paper
  • 149 Downloads

Abstract

The geomechanical characterization of a carbonate reservoir is required for formation stimulation and hydrocarbon recovery. The pertinent core- or block-scale (large-scale) characterizations are time consuming and expensive, and more importantly, cannot be used for drill cuttings. The present study proposes a two-scale model based on microscale (small-scale) measurements to predict the geomechanical properties of a carbonate formation at the core scale. At the small scale, we develop a physically representative element by accounting for the effective stiffness of a constitutive mineral and of voids. At the large scale, we account for the volume fraction of each mineral, the porosity, and the pore structure of the void space. The elastic deformation of a large-scale model is simulated using a finite element method (FEM), whose results are tested against independent lab measurements. The proposed two-scale model has applications for geomechanical characterization of a formation at the core scale from drill cuttings.

Keywords

Two-scale model Representative element Pore structure Finite element method (FEM) Drill cutting 

List of Symbols

\({A_{\text{m}}}\)

Cross-sectional area of the grain with a known mineralogy

\({A_{\text{M}}}\)

Cross-sectional area of the core-scale model

\({A_{\text{r}}}\)

Cross-sectional area of the representative element

Em

Elastic modulus of the mineral

\({E_{\text{M}}}\)

Elastic modulus of the core-scale model

\(E_{{{\text{Model}}}}^{{{\text{Ave}}}}\)

Average of the predicted elastic moduli

\(E_{{{\text{Lab}}}}^{{{\text{Max}}}}\)

Maximum of the measured elastic moduli

\(E_{{{\text{Lab}}}}^{{{\text{Min}}}}\)

Minimum of the measured elastic moduli

\({E_{\text{r}}}\)

Elastic modulus of the representative element

Error

Error of the predicted Young’s moduli at the core scale

\({f_i}\)

Volume fraction of each mineral

\({G_{\text{m}}}\)

Shear modulus

\({I_{\text{m}}}\)

Moment of inertia

\({J_{\text{m}}}\)

Polar moment of inertia

\({k_{\text{r}}}\)

Stretching stiffness

\({k_\theta }\)

Bending stiffness

\({k_\phi }\)

Torsional stiffness

\({L_{\text{m}}}\)

Average size of a solid grain with a known mineralogy

\({L_{\text{M}}}\)

Length of the core-scale model

\({L_{\text{r}}}\)

Length of the representative model

M

Bending load

N

Axial load

\({N_i}\)

Number of the representative elements relevant to the ith mineral in the core-scale model

\({N_{\text{t}}}\)

Total number of the representative elements in the core-scale model

P

Compressive load

T

Torsion of a solid medium

\({U_{\text{A}}}\)

Stretching or compression potential energy

\({U_{\text{M}}}\)

Bending potential energy

\({U_{\text{r}}}\)

Stretching energy of a solid medium

\({U_{\text{T}}}\)

Torsional energy

\({U_{{\text{total}}}}\)

Total potential energy of the solid medium

\({U_\theta }\)

Angle bending energy of the solid medium

\({U_\phi }\)

Torsional energy of the solid medium

\(\alpha\)

Rotational angle of the solid medium ends

\(\varepsilon\)

Normal strain of the model

\(\Delta {L_{\text{m}}}\)

Change in the length of the solid medium

\(\Delta r\)

Stretching elastic deformation

\(\Delta \beta\)

Torsion angle of the solid medium

\(\Delta \theta\)

Angle bending elastic deformation

\(\Delta \phi\)

Torsional elastic deformation

\(\sigma\)

Normal stress of the model

\({\phi _1}\)

Microporosity

\({\phi _2}\)

Macroporosity

\({\phi _{{\text{total}}}}\)

Total porosity

Abbreviations

FEM

Finite element method

Micro-CT

Micro computed tomography

XRD

X-ray diffraction

Notes

Acknowledgements

We are grateful for the constructive comments of the anonymous reviewers and the editor, which helped us improve the paper substantially.

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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Petroleum EngineeringUniversity of HoustonHoustonUSA

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