Pressure dependence of elastic deformation at multiple contacts and estimation of contact state of thin cracks

The pressure dependence of elastic deformation at multiple contacts in thin cracks has been investigated through the variation of elastic wave velocity. By extending previously introduced single-contact load-displacement relationships to those of multiple contacts and obtaining the first derivative of pressure with respect to the displacement at contacts, relationships for the pressure dependence of elastic deformation are obtained. The obtained relationships indicated that the pressure dependence is given by a pressure exponent μ, which is the multiple contact state of thin cracks. This was correlated with a similar pressure exponent in an empirically derived elastic wave velocity relationship, and we found that the contact state can be obtained by the empirical relationship. The estimated multiple contact state has two ideal values of 0.67 and 0.5 for conical and flat contacts, respectively, and μ values between the ideal states give the pressure dependence of elastic deformation of contacts, formed by a mixture of two ideal states and their population. The contact state values smaller than 0.5 and values greater than 0.67 indicate closed and open state of cracks, respectively. From previously published data on elastic wave velocities of various rocks, it is verified that the pressure dependence of elastic deformation of multiple contacts can be quantitatively obtained by the estimated contact states from the pressure dependence of elastic wave velocity.