Coffee is one of the most widely consumed beverages in the world. Making brewed coffee is an established art and business, practiced throughout the world. This fame of coffee has stimulated extensive coffee-oriented scientific research, an important component of which is in-depth knowledge of the physicochemical processes occurring in the coffee preparation. The extraction of espresso coffee can be described as a percolation process that exploits fluid-dynamic components. We consider a model that describes the water percolation problem in a porous medium. In more detail, it consists of the dynamics of the water flow, the diffusion and transport of the relevant soluble species and solid fine particles. From the knowledge of physical parameters of the extraction process and the concentration of relevant chemical compounds in the coffee powder, this model computes the hydraulic head, fluid flux, temperature of the pod during the percolation process, as well as the chemical characteristics of the espresso coffee in terms of the final concentration of the relevant compounds in the coffee beverage. The corresponding problem is an initial-boundary-value problem, on a cylindrical spatial domain. It undergoes a semi-discrete approximation through the collocation method, using Radial Basis Functions as the basis for the solution spatial approximation. Results obtained through numerical experiments and extraction campaigns supported by chemical laboratory analyses will be presented.