Anisotropic effective‐medium modeling of the elastic properties of shales

Shales are complex porous materials, normally consisting of percolating and interpenetrating fluid and solid phases. The solid phase is generally comprised of several mineral components and forms an intricate and anisotropic microstructure. The shape, orientation, and connection of the two phases control the anisotropic elastic properties of the composite solid. We develop a theoretical framework that allows us to predict the effective elastic properties of shales. Its usefulness is demonstrated with numerical modeling and by comparison with established ultrasonic laboratory experiments. The theory is based on a combination of anisotropic formulations of the self‐consistent (SCA) and differential effective‐medium (DEM) approximations. This combination guarantees that both the fluid and solid phases percolate at all porosities. Our modeling of the elastic properties of shales proceeds in four steps. First, we consider the case of an aligned biconnected clay‐fluid composite composed of ellipsoidal inclusion...

[1]  Donald F. Winterstein,et al.  Velocity anisotropy in shale determined from crosshole seismic and vertical seismic profile data , 1990 .

[2]  J. Berryman,et al.  Exact results for generalized Gassmann's equations in composite porous media with two constituents , 1991 .

[3]  D. H. Johnston.  Physical properties of shale at temperature and pressure , 1987 .

[4]  Predicting the Overall Properties of Composite Materials with Small-scale Inclusions or Cracks , 1989 .

[5]  A. Callegari,et al.  Consistent theoretical description for electrical and acoustic properties of sedimentary rocks , 1984 .

[6]  R. Roe.  Description of Crystallite Orientation in Polycrystalline Materials. III. General Solution to Pole Figure Inversion , 1965 .

[7]  J. D. Eshelby.  The determination of the elastic field of an ellipsoidal inclusion, and related problems , 1957, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[8]  Sheng.  Effective-medium theory of sedimentary rocks. , 1990, Physical review. B, Condensed matter.

[9]  R. Hill.  The Elastic Behaviour of a Crystalline Aggregate , 1952 .

[10]  S. Shtrikman,et al.  A variational approach to the theory of the elastic behaviour of multiphase materials , 1963 .

[11]  D. A. G. Bruggeman.  Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen , 1935 .

[12]  James G. Berryman,et al.  Long‐wavelength propagation in composite elastic media I. Spherical inclusions , 1980 .

[13]  N. Banik.  Velocity anisotropy of shales and depth estimation in the North Sea basin , 1984 .

[14]  R. Courant,et al.  Methods of Mathematical Physics , 1962 .

[15]  C. Sayers.  Inversion of ultrasonic wave velocity measurements to obtain the microcrack orientation distribution function in rocks , 1988 .

[16]  Herbert F. Wang,et al.  Ultrasonic velocities in Cretaceous shales from the Williston basin , 1981 .

[17]  B. Budiansky.  On the elastic moduli of some heterogeneous materials , 1965 .

[18]  J. Willis.  Bounds and self-consistent estimates for the overall properties of anisotropic composites , 1977 .