Inverse problem in anisotropic poroelasticity: drained constants from undrained ultrasound measurements

Poroelastic analysis has traditionally focused on the relationship between dry and drained constants, which are assumed known, and the saturated or undrained constants, which are assumed unknown. However, there are many applications in this field of study for which the main measurements can only be made on the saturated/undrained system, and then it is uncertain what the effects of the fluids were on the system, since the drained constants remain a mystery. The work presented here shows how to deduce drained constants from undrained constants for anisotropic systems having symmetries ranging from isotropic to orthotropic. Laboratory ultrasound data are then inverted for the drained constants in three granular packings: one of glass beads, and two others for distinct types of more or less angular sand grain packings. Experiments were performed under uniaxial stress, which resulted in hexagonal (transversely isotropic) symmetry of the poroelastic response. One important conclusion from the general analysis is that the drained constants are uniquely related to the undrained constants, assuming that porosity, grain bulk modulus, and pore fluid bulk modulus are already known. Since the resulting system of equations for all the drained constants is linear, measurement error in undrained constants also propagates linearly into the computed drained constants.