A Matrix Analysis of Operator-Based Upscaling for the Wave Equation

Scientists and engineers who wish to understand the earth’s subsurface are faced with a daunting challenge. Features of interest range from the microscale (centimeters) to the macroscale (hundreds of kilometers). It is unlikely that computational power limitations will ever allow modeling of this level of detail. Numerical upscaling is one technique intended to reduce this computational burden. The operator-based algorithm (developed originally for elliptic flow problems) is modified for the acoustic wave equation. With the wave equation written as a first-order system in space, we solve for pressure and its gradient (acceleration). The upscaling technique relies on decomposing the solution space into coarse and fine components. Operator-based upscaling applied to the acoustic wave equation proceeds in two steps. Step one involves solving for fine-grid features internal to coarse blocks. This stage can be solved quickly via a well-chosen set of coarse-grid boundary conditions. Each coarse problem is solved independently of its neighbors. In step two we augment the coarsescale problem via this internal subgrid information. Unfortunately, the complexity of the numerical upscaling algorithm has always obscured the physical meaning of the resulting solution. Via a detailed matrix analysis, the coarse-scale acceleration is shown to be the solution of the original constitutive equation with input density field corresponding to an averaged density along coarse block edges. The pressure equation corresponds to the standard acoustic wave equation at nodes internal to coarse blocks. However, along coarse cell boundaries, the upscaled solution solves a modified wave equation which includes a mixed second-derivative term.