Pre-Darcy flow in tight and shale formations

There are evidences that the fluid flow in tight and shale formations does not follow Darcy law, which is identified as pre-Darcy flow. Here, the unsteady linear flow of a slightly compressible fluid under the action of pre-Darcy flow is modeled and a generalized Boltzmann transformation technique is used to solve the corresponding highly nonlinear diffusivity equation analytically. The effect of pre-Darcy flow on the pressure diffusion in a homogenous formation is studied in terms of the nonlinear exponent, $m$, and the threshold pressure gradient, $G_{\mathrm{1}}$. In addition, the pressure gradient, flux, and cumulative production per unit area for different $m$ and $G_{\mathrm{1}}$ are compared with the classical solution of the diffusivity equation based on Darcy flow.