Mapping of hydraulic fractures from tiltmeter measurements

In considering the problem of inverse modeling of tiltmeter data for hydraulic fracture mapping, we address the issues of selecting the elastic model to represent the hydraulic fracture and limitations imposed by distance and fracture size on the information that can be recovered about the fracture. A tiltmeter measures, at its location, the changes in the surface inclination in two orthogonal directions. These inclinations are a direct measure of the horizontal gradient of the vertical component of the displacement field. Since advances in instrumentation in the last two decades, this type of apparatus have become extremely precise and can detect inclination changes down to a nanoradian. The simplicity of tiltmeter measurements has attracted interest not only in geophysics, but also in the petroleum industry. The idea of using tiltmeters to monitor hydraulic fractures can be traced back to the paper of Sun te{S} and is now a commercial service offered to the petroleum industry te{W}. However, the modeling and associated inverse problems required to analyze tiltmeter data raise difficult questions. The object(s) (fault, dyke, fracture) responsible for the recorded tilt are often modeled by finite Displacement Discontinuities, also called dislocation models. The validity of this type of model has been extensively discussed te{O,E} and many solutions for different configurations can be found in the literature. We show that it is possible to construct the solution for any type of dislocation model from the fundamental solution for an infinitesimal Displacement Discontinuity tensor. The eigenstrain theory te{M} is used to obtain this fundamental solution from the Green's function for the desired elastic domain (e.g. full or half space). Comparisons with known solutions demonstrate the flexibility of such method. We then focus on the problem of obtaining information about the orientation and size of an opening mode hydraulic fracture from the measured tilt field. One important problem is the identification of all the dimensions of the fracture model (length, width). The ability to obtain these parameters is controlled by limits, expressed in terms of the distance between the measurements and the fracture compared to the size of the fracture itself. The value of this ratio provides a condition that must be met before the fracture length-scales can be resolved. Determination of the fracture orientation is then investigated using a spatial Fourier Transform on the data set. This procedure highlights the requirement on the measurement array needed for a reliable identification: extension, number of tiltmeters, relative angle between the array and the fracture plane.