Inverse source problems for focusing wave energy to targeted subsurface formations: theory and numerical experiments
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Economically competitive and reliable methods for the removal of oil or contaminant particles from the pores of geological formations play a crucial role in petroleum engineering, hydro-geology, and environmental engineering. Post-earthquake observations at depleted oil fields as well as limited field experiments suggest that stress wave stimulation of a formation may lead to the release of particles trapped in its interstices. The stimulation can be applied using wave sources placed on or below the ground surface, and, typically, the effectiveness of the stimulation is proportional to the magnitude of the wave motion generated in the geological formation of interest. When wave sources are used to initiate the wave motion, equipment limitations and various sources of attenuation impose restrictions on the magnitude of the wave motion induced in the target formation. Thus, the engineering design of the wave energy delivery systems that are able to produce the wave motion of a required magnitude in the target zone is key to a successful mobilization of trapped interstitial particles. In this work, we discuss an inverse source approach that yields the optimal source time signals and source locations and could be used to design wave energy delivery systems. We cast the underlying forward wave propagation problem in two or three spatial dimensions. We model the target formation as an elastic or poroelastic inclusion embedded within heterogeneous, elastic, semi-infinite hosts. To simulate the semi-infiniteness of the elastic host, we augment the (finite) computational domain with a buffer of perfectly-matched-layers (PMLs). We define a metric of the wave motion generated in the target inclusion to quantify the amount of the delivered wave energy. The inverse source algorithm is based on a systematic framework of constrained optimization, where minimization of a suitably defined objective functional is tantamount to the maximization of the motion metric of the target formation. We demonstrate, via numerical experiments, that the algorithm is capable of converging to the spatial and temporal characteristics of surface loads that maximize energy delivery to the target formation. The numerical-simulation-based methodology is based on the assumption of perfect knowledge of the material properties and of the overall geometry of the geostructure of interest. In practice, however, precise knowledge of the properties of the geological formations is elusive, and quantification of the reliability of a deterministic approach is crucial for evaluating the technical and economical feasibility of the design. To this end, we also discuss a methodology that could be used to quantify the uncertainty in the wave energy delivery. Specifically, we treat the material properties of the layers as random variables, and perform a first-order uncertainty analysis of the elastodynamic system to compute the probabilities of failure to achieve threshold values of the motion metric. We illustrate the uncertainty quantification procedure for the case of two-dimensional, layered, isotropic, elastic host containing an elastic target inclusion. The inverse source and the uncertainty quantification methodologies, in conjunction, can be used for designing the characteristics of the wave sources used to deliver the wave energy to a targeted subsurface formation.