Summary
A novel method to map asymmetric hydraulic-fracture propagation using tiltmeter measurements is presented. Hydraulic fracturing is primarily used for oil-and-gas well stimulation, and is also applied to precondition rock before mining. The geometry of the developing fracture is often remotely monitored with tiltmeters—instruments that are able to remotely measure the fracture-induced deformations. However, conventional analysis of tiltmeter data is limited to determining the fracture orientation and volume. The objective of this work is to detect asymmetric fracture growth during a hydraulic-fracturing treatment, which will yield height-growth information for vertical fracture growth and horizontal asymmetry for lateral fracture growth or detect low preconditioning-treatment efficiency in mining. The technique proposed here uses the extended Kalman filter (EKF) to assimilate tilt data into a hydraulic-fracture model to track the geometry of the fracture front. The EKF uses the implicit level set algorithm (ILSA) as the dynamic model to locate the boundary of the fracture by solving the coupled fluid-flow/fracture-propagation equations, and uses the Okada half-space solution as the observation model (forward model) to relate the fracture geometry to the measured tilts. The 3D fracture model uses the Okada analytical expressions for the displacements and tilts caused by piecewise constant-displacement discontinuity elements to discretize the fracture area. The proposed technique is first validated by a numerical example in which synthetic tilt data are generated by assuming a confining-stress gradient to generate asymmetric fracture growth. The inversion is carried in a two-step process in which the fracture dip and dip direction are first obtained with an elliptical fracture-forward model, and then the ILSA-EKF model is used to obtain the fracture footprint by fixing the dip and dip direction to the values obtained in the first step. Finally, the ILSA-EKF scheme is used to predict the fracture width and geometry evolution from real field data, which are compared with intersection data obtained by temperature and pressure monitoring in offset boreholes. The results show that the procedure is able to satisfactorily capture fracture growth and asymmetry even though the field data contain significant noise, the tiltmeters are relatively far from the fracture, and the dynamic model contains significant “unmodeled dynamics” such as stress anisotropy, material heterogeneity, fluid leakoff into the formation, and other physical processes that have not been explicitly accounted for in the dynamic ILSA model. However, all the physical processes that affect the tilt signal are incorporated by the EKF when the tilt measurements are used to obtain the maximum likelihood estimates of the fracture widths and geometry.
Load Parameter Identification for Parallel Robot Manipulator Based on Extended Kalman Filter
