On the Limitations of Effective Medium Approaches for Estimating Fracture Width in Anisotropic Rocks

It is always tempting to be able to replace a complex problem requiring characterization of multiple input parameters with a simplified one that requires reduced number of inputs and takes less effort and time to develop and then apply the solution procedures to obtain an approximate solution reasonably close to the true one. During the modeling of heterogeneous geologic rock masses, in which the main variability is typically realized as horizontal lamination at different scales, this approach is often implemented as homogenization or upscaling of multiple adjacent layers into fewer and thicker layers with equivalent parameters or responses. Even with the most powerful computational resources, numerical simulations of rocks require that real variability of rock properties be smoothed to the level that can be reproduced on the numerical mesh with the highest refinement. This work quantifies how large the worst-case scenario errors can be when the homogenization approach is applied to a particular problem of estimating the width of a fracture in a layered formation in cases of both isotropic and anisotropic elastic layers.