Continuous upscaling of the 3D diffusion equation in a heterogeneous medium

We describe the procedure of continuous upscaling of the flow equations. Multiscale averaging is a necessary operation in many applications. Under the introduction of the continuous line of scales, upscaling becomes similar to a Markov process described by a partial differential equation of the Ornstein-Uhlenbeck type. While previously [Shapiro, Chem. Eng. Sci. 234, 116454] the procedure of continuous upscaling was formulated for 1D flows, here we generalize it onto 3D processes. For upscaling 3D fluxes, new laws are formulated. The procedure is applied to upscale the steady-state diffusion (or heat conduction, or the pressure equation) in a heterogeneous medium. Rules for upscaling the diffusion coefficient are derived. In many cases, the upscaling of the diffusion coefficient may depend on the chosen class of solutions. It is studied numerically. The lowest values of the coefficient and the zones of its sharp variation contribute most to upscaling.