Numerical modelling of seismic waves scattered by hydrofractures: application of the indirect boundary element method

We use a numerical method that can model the seismic waveforms scattered from an arbitrary number of fractures that are either empty, or contain elastic or fluid material. The indirect boundary element method (BEM) is capable of generating the full elastic wavefield and is programmed in 2-D. The governing equations and discrete implementation of the technique are described. We explain in detail a new approach for evaluating the improper boundary integrals. The method is shown to be highly accurate from a comparison with mode summation. Subsequently, the BEM is applied to modelling hydrofractures. Synthetic examples, calculated for cross-well and single well geometries, demonstrate the effects of crack length, opening and infill on recorded displacements. It is shown that diffractions from the tips can, in principle, be used to locate and determine the hydrofracture size. These arrivals depart from ray theoretical travel-times due to defocusing over a Fresnel zone. S-wave diffractions generally have a larger amplitude than the diffracted P-waves, and so may provide a better indication of fracture size. Energy which is converted to interface waves and subsequently diffracted from the crack tips is also observed. The presence of water allows energy to pass through the fracture; this is clearly evident on the cross-well seismograms. Closure of the fracture causes a further increase in transmission amplitude as less energy is attenuated through internal multiples.