Optimization of sources for focusing wave energy in targeted formations

Abstract

We discuss a numerical approach for identifying the surface excitation that is necessary to maximize the response of a targeted subsurface formation. The motivation stems from observations in the aftermath of earthquakes, and from limited field experiments, whereby increased oil production rates were recorded and were solely attributable to the induced reservoir shaking. The observations suggest that focusing wave energy to the reservoir could serve as an effective low-cost enhanced oil recovery method. In this paper, we report on a general method that allows the determination of the source excitation, when provided with a desired maximization outcome at the targeted formation. We discuss, for example, how to construct the excitation that will maximize the kinetic energy in the target zone, while keeping silent the neighbouring zones. To this end, we cast the problem as an inverse-source problem, and use a partial-differential-equation-constrained optimization approach to arrive at an optimized source signal. We seek to satisfy stationarity of an augmented functional, which formally leads to a triplet of state, adjoint and control problems. We use finite elements to resolve the state and adjoint problems, and an iterative scheme to satisfy the control problem to converge to the sought source signal. We report on one-dimensional numerical experiments in the time domain involving a layered medium of semi-infinite extent. The numerical results show that the targeted formation's kinetic energy resulting from an optimized wave source could be several times greater than the one resulting from a blind source choice, and could overcome the mobility threshold of entrapped reservoir oil.