A True Poroelastic Up and Downscaling Scheme for Multi-scale Coupled Simulation

Multi-scale simulation allows an efficient use of computational resources when one to solve problems that have domains with very different spatial scales and can have significantly different discretizations. This is often the case in coupled reservoir flow-geomechanics simulations. Because of the fluid rock interaction, multi-scale simulation in this context requires a consistent upscaling of pore pressure and downscaling of pore volume change.

In general, upscaling of the pore pressure will happen over a collection of cells where the change in pressure is not constant. This is to be expected due to diffusive propagation of pressure perturbations in the system. The differences in pressure perturbations will also be enhanced by heterogeneities in porosity or permeability. The impact of heterogeneity in flow properties is also exacerbated when considering variations in the mechanical properties. Clearly any type of upscaling scheme must consider the combination of pressure and mechanical properties at the fine scale to determine their effective properties at the coarse scale. The same is true if you want to downscale pore volume calculated at the coarse scale to the fine scale.

Here a solution to this problem is presented which extends the Backus upscaling scheme using the pore pressure – apparent porosity thermodynamic conjugate. A solution is also presented for downscaling the pore volume change from the equivalent anisotropic medium at the coarse scale to the layered, fine scale medium. Validation of this method is given by consideration of a layered Mandel's problem. The behavior of numerical simulations is shown to be in good agreement with the analytical result using the proposed upscaling method.