Microcrack connectivity in rocks: A renormalization group approach to the critical phenomena of conduction and failure in crystalline rocks

A study is made of the critical phenomena associated with the onset of conductivity and the onset of failure in a rock with a random distribution of microcracks using a renormalization group theory approach. The interaction between different length scales is an important element in the theory, which is a departure from the results of equivalent media theory. The crack parameter that determines if the rock can conduct or is to fail is the crack count per unit area times the square of the crack length. The parameter space is divided up into regions or phases. These phases are those of no conduction and no failure, conduction and no failure, and conduction and failure. The predictions of the model agree well with SEM data on the microcrack populations of stressed and unstressed rock samples. The crack densities seen in unstressed Westerly granite fall in the range that predicts conduction but no failure, but only a 50% increase in density is needed to reach the failure criteria. A sample stressed to 65% of the failure stress has greatly increased conduction connectivity but seems no closer to failure. The sample stressed to failure shows an appropriate crack density increase.