A unified asperity-deformation model for cracked rocks

Seismic velocities in rocks increase with pressure, a pattern often explained by changes in cracks within the rock volume. Specifically, the width of microcracks will decrease, increasing the contact area of crack surfaces, which in turn leads to increased stiffness of the crack. This causes P- and S-wave velocities to increase, and having accurate models of this behavior is important in many applications where it is important to related stress changes and observed seismic velocities, such as investigations of both fault zones and changing conditions in geothermal fields or hydrocarbon reservoirs. Several different models have been proposed to explain and quantify the relationship between confining pressure and the physical properties of cracked rocks, including a number of solutions describing cracks as ellipsoidal voids (penny-shaped cracks) of with varying aspect ratios that close at different pressures. Differential effective medium theories based on this model will typically use a fairly large number of parameters to fit measured data. An alternative approach describes cracks or fractures as rough surfaces that do not fully close with increasing pressure, and estimates the increase in contact area as asperities come into contact. An asperity-deformation model of this type was developed by, e.g., Gangi and Carlson (1985), and it parameterizes asperities on crack surfaces in terms of a set of cylindrical rods with heights following power-law distribution. The model results in a simple expression for the increase in velocity caused by increased number of asperities (rods) in contact with increasing pressure, an expression that accurately models increases in velocity using only three parameters. However, the model was formulated only for normal modulus of the cracks, the modulus relevant for P-wave propagation across the crack. Since S-waves are controlled by the tangential or shear modulus of the crack, this model cannot be used to jointly invert both P- and S-wave velocities measurements for a single model of the asperity height distribution that predicts the rock behavior. We therefore extend the previous model to consider also the shear stress exerted on the crack, which will cause the rods to bend laterally. Using Euler-Bernoulli beam theory, we relate the power-law asperity height distribution to the shear compliance of the crack as well as the normal compliance. One additional parameter describing this lateral deformation of asperities is included in the new relation. The result shows that the value of the shear compliance is determined by normal pressure on the crack but not by the shear pressure, although the shear pressure causes the lateral deformation of asperities. This indicates that changes in normal and shear compliances of the crack are not independent. We use the model to express the pressure dependence of P- and S-wave velocities, and apply these solutions to inverse nine sets of published data from laboratory measurements of P- and S-wave velocities as a function of confining pressure for dry rock samples. The results show that the model can fit both types of velocity measurements very accurately over a pressure range of about 100 MPa using only four parameters. Ongoing work aims to improve inversion methods and to extend the model to include the influence of fluids.