Inverse modeling for estimating fluid-overpressure distributions and stress intensity factors from an arbitrary open-fracture geometry

We present a new solution for estimating the fluid overpressure (driving-pressure or net-pressure) acting on the walls of a fracture with an arbitrary opening displacement. In the paper, we first present a forward modeling solution, using Fourier cosine series, for the opening displacement of a fracture subject to an overpressure that varies irregularly along the length of a fracture. By changing the form of the solution, we provide a matrix equation for estimating the Fourier coefficients and thereby obtain the overpressure variation from the fracture geometry. As numerical tests of this inverse analysis, we estimated the overpressure variation from fracture-opening displacements given by well known analytical solutions, and found that this method can be used for overpressure estimates for a variety of fluid-driven fractures. We apply our solutions to a mineral vein hosted by gneiss (West Norway) and conclude from the aperture variation that, at the time of vein formation, the overpressure increased toward the vein tip. We also discuss the physical meaning of the Fourier coefficients by applying our results to man-made hydraulic wing fractures. The results indicate that the coefficients of n = 0 and n = 1 relate to the fluid overpressure and the critical stress intensity factor.