Non-linear elastic behaviour of damaged rocks

The pervasive damage of rocks by microcracks and voids strongly affects their macroscopic elastic properties. To evaluate the damage effects, we derive here the macroscopic stress-strain relations for a 3-D elastic solid with non-interacting cracks embedded inside a homogeneous matrix. The cracks considered are oriented either perpendicular to the maximum tension axis, or perpendicular to the maximum compression axis. In the first case they dilate during loading and in the second case they contract during loading. We derive a solution for the elastic energy of this rock following the self-consistent scheme of Budiansky & O'Connell (1976). The solution describes the stress-strain relations in terms of Λ^d and μ^d, which are the modified Lame constants, and an additional parameter Λ. The latter accounts for the non-linear behaviour of the solid and is related to crack density. The solution predicts a non-linear elastic rheology even for an infinitesimal strain of ɛ < 0.001, and abrupt change in the elastic moduli when the loading reverses from uniaxial compression to uniaxial tension.
We use the derived solution to analyse rock-mechanics experiments in which beams of Indiana limestone were deformed under four-point loading. This configuration provides simultaneously the apparent tensile and compressive moduli for small strains. The apparent moduli fit the effective elastic moduli calculated with the present damage model well, including the differences between tensile and compressive moduli.     

Attenuation behaviour of tuffaceous sandstone and granite during microfracturing

Laboratory measurements of ultrasonic wave propagation in tuffaceous sandstone (Kimachi, Japan) and granite (Iidate, Japan) were performed during increasing fracturing of the samples. The fracturing was achieved by unconfined uniaxial compression up to and beyond the point of macrofracture of the specimen using a constant low strain rate. The observed variation of wave velocity (up to 40 per cent) due to the development of micro- and macrofractures in the rock is interpreted by rock models relating velocity changes to damage and crack density. The calculated density of the newly formed cracks reaches higher values for the sandstone than for the granite. Using the estimated crack densities, the attenuation behaviour is interpreted in terms of different attenuation mechanisms; that is, friction and scattering. Rayleigh scattering as described by the model of Hudson (1981 ) may explain the attenuation qualitatively if the largest plausible crack dimensions are assumed in modelling.     

Time-lapse traveltime change of singly scattered acoustic waves

We present a technique based on the single-scattering approximation that relates time-lapse localized changes in the propagation velocity to changes in the traveltime of singly scattered waves. We describe wave propagation in a random medium with homogeneous statistical properties as a single-scattering process where the fluctuations of the velocity with respect to the background velocity are assumed to be weak. This corresponds to one of two end-member regimes of wave propagation in a random medium, the first being single scattering, and the second multiple scattering. We present a formulation that relates the change in the traveltime of the scattered waves to a localized change in the propagation velocity by means of the Born approximation for the scattered wavefield. We validate the methodology with synthetic seismograms calculated with finite differences for 2-D acoustic waves. Potential applications of this technique include non-destructive evaluation of heterogeneous materials and time-lapse monitoring of heterogeneous reservoirs.