The impact of Brinkmanʹs extension of Darcyʹs law in the neighborhood of a circular preferential flow pathway

Flow through preferential pathways can be a significant contribution to the total flow through a porous medium. Analytical strategies for specifying conditions on preferential pathway flow boundaries are considered. A new analytical solution is presented based on Brinkmanʹs extension of Darcyʹ s law for flow in and around a preferential pathway through a porous medium. Nondimensional analysis is presented to quantify the solutionʹs predicted impact compared to solutions based on Darcyʹs law as the pathway boundary condition. Analysis is also presented to quantify the magnitude of essential parameters required to produce significant results. These results indicate that for some parameter values, Brinkmanʹs extension of Darcyʹs law predicts a significantly higher volumetric flux within the preferential pathway, and predicts that velocities in excess of the Darcy velocity will be induced for a significant distance into the porous medium. There has been a struggle to develop appropriate boundary conditions for this problem. Brinkmanʹs extension of Darcyʹs law is clearly not an ideal solution. However, the approach has an interesting experimental and analytical history, has conceptual merit, responds in a way that is intuitively plausibility, and provides predictions that can be tested experimentally. It is hoped that presenting this new analytical solution will help to advance the scientific discourse on this subject and may provide insights that lead to new kinds of physical experimentation.