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Monitoring Evolving Hydraulic Fracture Growth using Tiltmeters and a combined Extended Kalman Filter-Implicit Level Set Algorithm

Banff International Research Station for Mathematical Innovation and Discovery

The inversion of remote tilt measurements to determine the geometry of an evolving hydraulic fracture (HF) is a classically ill-posed problem due to the fact that the elliptic PDE that governs the behaviour of the displacement gradient field rapidly smooths the geometric details of the fracture with distance. Indeed, it is possible to show that it is only possible to obtain reasonable estimates of the first few moments of the crack opening displacement field from such measurements. On the other hand, numerical and analytic models of evolving HF cannot be expected to provide a completely accurate prediction of evolving HF geometries taking place in the field due to the large number of uncertainties in the data and the un-modeled dynamics due to physical processes that have, of necessity, been ignored. Our approach is to feed the time series of tilt data as input to the implicit level set algorithm (ILSA) model for an evolving HF via the Extended Kalman Filter (EKF). Form an inversion point of view, the dynamics from the coupled ILSA model enables the tilt data snapshots in the time series to be connected, where in previous inversion algorithms these data were regarded as independent. We illustrate the EKF-ILSA algorithm using numerical experiments for planar HF propagating in 3D elastic media. By varying the confining stress field, synthetic tiltmeter data are generated that result in substantial changes to the geometry of the evolving HF. The ILSA model is assumed to have no knowledge of this confining stress field except for feedback from the tiltmeters via the Extended Kalman Filter. Indeed, without this feedback the ILSA HF model would propagate with radial symmetry. We compare the EKF-ILSA estimates of the fracture geometry and width with those of the HF used to generate the synthetic data with and without Gaussian noise. We also present results in which the algorithm is tested on real field data from a mining situation in which HF have been deliberately generated to enhance caving in longwall coal mining. The model is able to detect asymmetry in the growth of the HF, which is corroborated by measured intersection times of the HF with monitoring boreholes.

Mathematics; Mechanics of deformable solids; Fluid mechanics

video/mp4

Author affiliation: University of British Columbia

2019-03-19

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/68918

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/