Poroelastic measurements resulting in complete data sets for granular and other anisotropic porous media

INTRODUCTION

Poroelastic analysis (Wang, 2000; Thigpen and Berryman, 1985; Gassmann, 1951; Rice and Cleary, 1976; Brown and Korringa, 1975; Biot and Willis, 1957; Cheng, 1997) usually progresses from assumed knowledge of dry or drained porous media to the predicted behavior of fluid-saturated and undrained porous media. When field data (say from oil, gas, or hydrological reservoirs) are involved, the experimental situation is generally and unfortunately incompatible with these assumptions.

For this reason, I want to consider several different experimental scenarios typified by one in which a set of undrained constants has been measured using either ultrasound (in the laboratory) or seismic wave analysis (for field data), while some or all of the dry or drained constants are usually unmeasured and therefore unknown. Drained constants for such a poroelastic system can be deduced from available data if a complete set of undrained compliance data (as could be calculated by inverting the stiffness data, which are almost directly obtained, within a factor of the density, from wave speed measurements) is available, together with a few other commonly measured quantities such as porosity, fluid bulk modulus, and grain bulk modulus for isotropic systems. Similar results are developed here for anisotropic systems having up to orthotropic symmetry if the system is granular (i.e., composed of solid grains assembled into solid either by a cementation process or by applied stress) and the grains are known to be elastically homogeneous.

In the later sections, the analysis is also fully developed for anisotropic systems with inhomogeneous (meaning more than one mineral type is present), but still isotropic, grains. Also studied is the case for uniform collections of the same types of anisotropic grains, as long as the grain symmetry axes are either perfectly aligned or perfectly random. I show how many poroelastic data are needed in order to consider the data sets complete, and which types of data are in some sense redundant. Some combinations can be used to replace other data types that remain missing when lab experimental and/or field limitations prevent direct measurements of all the poroelastic coefficients.

Poroelastic measurements resulting in complete data sets for granular and other anisotropic porous media