Cascade Inversion for Tilt Data

Abstract

We investigate an inverse problem for mapping hydraulic fracture geometry using surface tiltmeter data. It is well known that this problem becomes a non-linear inverse problem. If the inversion algorithm can be separated into two stages as: (1) the identification of the fracture plane and (2) the estimation of the fracture aperture distribution, the computation scheme becomes robust and stable. We call this methodology a cascade inversion scheme for tilt data. In the first stage of the inversion, the fracture plane is estimated by the Nelder-Mead simplex method whereby we obtain the volume of the hydraulic fracture. The fracture aperture distribution is then determined by the successive linear inversion stage. In this second stage, the fracture plane is divided into small rectangular pieces and each piece has a different fracture aperture. The normal equation, where the fracture apertures are unknowns, can be solved by the least squared method with two constraints, which are the smoothness and non-negative values of the fracture apertures. The proposed methodology was applied to synthetic and field data. The inversion results are quite acceptable and we conclude that this cascade inversion scheme is a robust method and easy to apply to field data.