Crack propagation models for rock fracture in a geothermal energy reservoir

The propagation of a one-dimensional, fluid-filled crack in a hot dry rock geothermal energy reservoir (HDRGER) is discussed. In previous studies a number of different relationships between the normal stress on the crack, the fluid pressure, and the crack height (so-called crack laws) have been used, as have different 'flow laws' to determine the relationship between flow rate and crack geometry. Here it is shown that the choice of submodel may have profound implications for the mathematical structure of the problem. In particular, two crack laws (a linear law and a hyperbolic law) are considered as well as two flow laws (a cubic law and a linear law). The model contains a dimensionless parameter that measures the relative importance of stresses due to local deformation of asperities and the long-range deformation of the crack surface. The case is considered where the former is the dominant mechanism. A perturbation analysis is performed, and it is found that for some combinations of laws a strained-coordinate analysis is required, whilst for others a matched asymptotic approach is needed. In the latter case the problem may be reduced to that of solving a linear, nonhomogeneous singular integrodifferential equation to determine the behaviour in the boundary layer. This problem is solved, and some conclusions are drawn regarding the relevance of various laws to flow in HDRGERs.