Modeling fluid flow in fractured porous media with a combined Phase-Field and Navier-Stokes technique

Numerical modeling of fluid flow in porous media with discontinuities often relies on simplifications that overlook important factors commonly encountered in real scenarios, such as interfacial conditions between fractures and the surrounding porous media, as well as deviations from classical flow theories, including Poiseuille flow. This includes fluid flow in two and three dimensions within cracks that have non-parallel walls, particularly at high Reynolds numbers. This paper introduces a novel set of continuous partial differential equations derived from the Navier-Stokes equations and Phase-Field theory to effectively model fluid flow through cracked, saturated porous media. The Phase-Field theory facilitates a continuous transition from the Navier-Stokes equations governing fluid flow in cracked domains to the mass and momentum conservation equations that govern intact porous media. The proposed model enhances the simulation accuracy of fluid flow through porous media that contain discontinuities, effectively addressing the complex conditions associated with cracks and discontinuities within the medium. The Characteristic-Based Split (CBS) method is employed to derive the governing equations. Furthermore, the computational model is validated against two benchmark problems within the CBS framework. The effectiveness of the proposed approach is demonstrated through two examples of fluid flow in cracked porous media; one including an inclined crack and the other presenting an edge crack with converging walls. The final example illustrates how channel tortuosity influences flow behavior in channels confined by porous media, highlighting flow patterns in channels with complex geometries.