We present new quantitative model describing the pressure dependence of acoustic P- and S-wave velocities. Assuming that a variety of individual mechanisms or defects (such as cracks, pore collapse and grain crushing) can contribute to the pressure-dependent change of the wave velocity, we order a characteristic pressure to all of them and allow a series of exponential terms in the description of the (P- and S-waves) velocity-pressure function. We estimate the parameters of the multi-exponential rock physical model in inversion procedures using laboratory measured P- and S-wave velocity data. As is known, the conventional damped least squares method gives acceptable results only when one or two individual mechanisms are assumed. Increasing the number of exponential terms leads to highly nonlinear ill-posed inverse problem. Due to this reason, we develop the spectral inversion method (SIM) in which the velocity amplitudes (the spectral lines in the characteristic pressure spectrum) are only considered as unknowns. The characteristic pressures (belonging to the velocity amplitudes) are excluded from the set of inversion unknowns, instead, they are defined in a set of fixed positions equidistantly distributed in the actual interval of the independent variable (pressure). Through this novel linear inversion method, we estimate the parameters of the multi-exponential rock physical model using laboratory measured P- and S-wave velocity data. The characteristic pressures are related to the closing pressures of cracks which are described by well-known rock mechanical relationships depending on the aspect ratio of elliptical cracks. This gives the possibility to estimate the aspect ratios in terms of the characteristic pressures.

Download PDF:

Keywords: Multi-exponential rock physical model, Spectral inversion method (SIM), Crack aspect ratio, Characteristic pressures