# Self-consistent imbedding and the ellipsoidal model model for porous rocks

## Abstract

Equations are obtained for the effective elastic moduli for a model of an isotropic, heterogeneous, porous medium. The mathematical model used for computation is abstract in that it is not simply a rigorous computation for a composite medium of some idealized geometry, although the computation contains individual steps which are just that. Both the solid part and pore space are represented by ellipsoidal or spherical 'grains' or 'pores' of various sizes and shapes. The strain of each grain, caused by external forces applied to the medium, is calculated in a self-consistent imbedding (SCI) approximation, which replaces the true surrounding of any given grain or pore by an isotropic medium defined by the effective moduli to be computed. The ellipsoidal nature of the shapes allows us to use Eshelby's theoretical treatment of a single ellipsoidal inclusion in an infiinte homogeneous medium. Results are compared with the literature, and discrepancies are found with all published accounts of this problem. Deviations from the work of Wu, of Walsh, and of O'Connell and Budiansky are attributed to a substitution made by these authors which though an identity for the exact quantities involved, is only approximate in the SCI calculation. This reduces the validity ofmore »

- Authors:

- Publication Date:

- Research Org.:
- Chevron Oil Field Research Co., Box 446, La Habra, California 90631

- OSTI Identifier:
- 5842833

- Resource Type:
- Journal Article

- Journal Name:
- J. Geophys. Res.; (United States)

- Additional Journal Information:
- Journal Volume: 84:B10

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 58 GEOSCIENCES; POROUS MATERIALS; ROCK MECHANICS; GRAIN DENSITY; POROSITY; PRESSURE DEPENDENCE; STRAIN RATE; CRYSTAL STRUCTURE; MATERIALS; MICROSTRUCTURE; 580300* - Mineralogy, Petrology, & Rock Mechanics- (-1989)

### Citation Formats

```
Korringa, J, Brown, R J.S., Thompson, D D, and Runge, R J.
```*Self-consistent imbedding and the ellipsoidal model model for porous rocks*. United States: N. p., 1979.
Web. doi:10.1029/JB084iB10p05591.

```
Korringa, J, Brown, R J.S., Thompson, D D, & Runge, R J.
```*Self-consistent imbedding and the ellipsoidal model model for porous rocks*. United States. https://doi.org/10.1029/JB084iB10p05591

```
Korringa, J, Brown, R J.S., Thompson, D D, and Runge, R J. Mon .
"Self-consistent imbedding and the ellipsoidal model model for porous rocks". United States. https://doi.org/10.1029/JB084iB10p05591.
```

```
@article{osti_5842833,
```

title = {Self-consistent imbedding and the ellipsoidal model model for porous rocks},

author = {Korringa, J and Brown, R J.S. and Thompson, D D and Runge, R J},

abstractNote = {Equations are obtained for the effective elastic moduli for a model of an isotropic, heterogeneous, porous medium. The mathematical model used for computation is abstract in that it is not simply a rigorous computation for a composite medium of some idealized geometry, although the computation contains individual steps which are just that. Both the solid part and pore space are represented by ellipsoidal or spherical 'grains' or 'pores' of various sizes and shapes. The strain of each grain, caused by external forces applied to the medium, is calculated in a self-consistent imbedding (SCI) approximation, which replaces the true surrounding of any given grain or pore by an isotropic medium defined by the effective moduli to be computed. The ellipsoidal nature of the shapes allows us to use Eshelby's theoretical treatment of a single ellipsoidal inclusion in an infiinte homogeneous medium. Results are compared with the literature, and discrepancies are found with all published accounts of this problem. Deviations from the work of Wu, of Walsh, and of O'Connell and Budiansky are attributed to a substitution made by these authors which though an identity for the exact quantities involved, is only approximate in the SCI calculation. This reduces the validity of the equations to first-order effects only. Differences with the results of Kuster and Toksoez are attributed to the fact that the computation of these authors is not self-consistent in the sense used here. A result seems to be the stiffening of the medium as if the pores are held apart. For spherical grains and pores, their calculated moduli are those given by the Hashin-Shtrikman upper bounds. Our calculation reproduces, in the case of spheres, an early result of Budiansky. An additional feature of our work is that the algebra is simpler than in earlier work. We also incorporate into the theory the possibility that fluid-filled pores are interconnected.},

doi = {10.1029/JB084iB10p05591},

url = {https://www.osti.gov/biblio/5842833},
journal = {J. Geophys. Res.; (United States)},

number = ,

volume = 84:B10,

place = {United States},

year = {1979},

month = {9}

}