Mechanics of stratified anisotropic poroelastic media

INTRODUCTION

Traditional poroelastic analysis (Thigpen and Berryman, 1985; Biot, 1962; Pride et al., 2002; Wang, 2000; Cheng, 1997; Rice and Cleary, 1976; Brown and Korringa, 1975; Zimmerman, 1991; Biot and Willis, 1957; Gassmann, 1951) usually progresses from assumed knowledge of dry or drained porous media to the predicted behavior of fluid-saturated and undrained porous media. This class of problems is characterized by a single upscaling step, taking the homogeneous fluid and solid constituent properties and deducing the macroscopic behavior of such systems. In recent work (Berryman, 2010), the author has shown in detail how the poroelastic coefficients are related to the microstructural constants of the solid constituents when the overall behavior varies from isotropic to orthotropic. The focus of the present work is on stratified (i.e., layered) poroelastic materials, which are therefore heterogeneous at the mesoscale but fairly homogeneous within each layer. In particular, individual layers are assumed to satisfy the same assumptions as the class of problems considered by Berryman (2010), which is basically limited to orthotropic poroelastic media with a known set of symmetry axes.

The main issue addressed here concerns how the interface boundary conditions between anisotropic porous layers should be treated. For very low frequency (say quasi-static) analysis, this issue is clear since then the boundary conditions must be drained conditions and therefore the fluid pressure is continuous across the boundary. However, for high frequency wave propagation, it is expected to be more appropriate to treat the system as locally undrained, since the pressure of the pore-fluid does not have time to equilibrate via the drainage mechanism, which can take much longer than is appropriate to these quasi-static analyses. The most accurate way to treat these situations is to consider the variables to be frequency dependent and complex. This approach has been taken for example by Pride et al. (2004); Pride and Berryman (2003b,a) for mixtures of isotropic poroelastic materials. But the problem becomes harder for the anisotropic case, as there were simple exact results for the two-isotropic-component case, but simple results are not available for the anisotropic problems. And more importantly, the interest in layered media is not just for two-component examples, but ultimately for multi-component layered media. So it is important to consider these cases separately, as is being done here.

The analysis is restricted to anisotropic systems. The nature of the grains themselves composing the solid frame material will not be a focus of the present paper. This issue does matter, but it is most important for determining the relationship between the grain constants and the off-diagonal coefficients that are called the $\beta$ 's in this formulation. These issues have been fully addressed in the earlier contribution of the author (Berryman, 2010), and will therefore not be treated again in this paper. Our focus here is on heterogeneous poroelastic media when the heterogeneity is well-represented via layered porous-medium modeling.

Mechanics of stratified anisotropic poroelastic media