Single-phase macroscopic flow in a rigid porous medium is traditionally described by classical Darcy's law which can be formally derived by upscaling the pore-scale flow equation in the creeping incompressible flow regime and the no-slip condition at the solid-fluid interfaces. However, there are many situations for which fluid release from the surface into the pores or, conversely, absorption from the pores into the solid through the interfaces may occur. To cite some but a few, this case is encountered in drying (Vu & Tsotsas, 2018) and pyrolysis (Mahmoudi et al., 2014) of porous materials, vapor bubble migration in ice due to temperature gradient (Shreve, 1967) or processes for which a chemical reaction in a porous material leads to a net production of fluid from the surface into the pores or, conversely, absorption from the pores into the solid through the interfaces. This translates into a local normal flux at the solid fluid-interface, featuring a generic problem which may be referred to as flow in exuding porous media. Although classical Darcy's law has been widely heuristically employed to describe this type of flow, the question remains on the physical relevance of such an assumption.