Einstein material balance and modeling of the flow of compressible fluid near the boundary

We consider sewing machinery between finite difference and analytical solutions defined at different scales: far away and near the source of the perturbation of the flow. One of the essences of the approach is that the coarse problem and the boundary-value problem in the proxy of the source model two different flows. In his remarkable paper, Peaceman proposes a framework for dealing with solutions defined on different scales for linear time independent problems by introducing the famous Peaceman well block radius. In this article, we consider a novel problem: how to solve this issue for transient flow generated by the compressibility of the fluid. We are proposing a method to glue solution via total fluxes, which are predefined on coarse grid, and changes in pressure, due to compressibility, in the block containing production (injection) well. It is important to mention that the coarse solution “does not see” the boundary. From an industrial point of view, our report provides a mathematical tool for the analytical interpretation of simulated data for compressible fluid flow around a well in a porous medium. It can be considered a mathematical “shirt” on the Peaceman well-block radius formula for linear (Darcy) transient flow but can be applied in much more general scenarios. In the article, we use the Einstein approach to derive the material balance equation, a key instrument to define $R_0$. We will expand the Einstein approach for three regimes of Darcy and non-Darcy flows for compressible fluids (time-dependent):
I. stationary;
II. pseudostationary;
III. boundary dominated.
Note that in all known authors literature, the rate of production on the well is time-independent.