We model porous media as two-dimensional networks of channels. As suspension flows through the network, particles clog channels. Assuming no flow through clogged channels, we determine an upper bound on the number of channels that clog and leave the network impervious. More precisely, the number of channels that clog does not exceed the number of faces of the multigraph associated with the network. In terms of d, the average coordination number of the network, the number of channels that clog does not exceed (1−2∕d) of the total number of channels.

Fluid in porous media flows through tortuous paths. If the size of the solid particles suspended in the fluid and the fluid velocities are large enough, Brownian motion effects may not be dominant. In this parameter regime, the average velocity of the particles is different than that of the fluid. We obtain the relation between the average velocity of the particles and the average velocity of the fluid in the context of a simple mathematical model and we apply our results to flows near wells.

Abstract.We model filters as two-dimensional networks of channels. As a suspension (fluid with particles) flows through the filter, particles clog channels. We assume that there is no flow through clogged channels. In this paper, we compute a sharp upper bound on the number of channels that can clog before fluid can no longer flow through the filter. Fluid suspensions (or suspensions, for short) are fluids with small solid particles in them. According to their size and properties, these particles are called fines or colloids. As a suspension flows through a permeable porous material, some fines are trapped within the material. In fact, the function of the filters we consider in this paper is to clean suspensions by capturing most particles bigger than a certain size.The removal of particles from fluid suspensions is of importance in a wide range of industrial and technological applications such as waste water treatment [18] and other filtration processes [4,31]. Our studies are motivated by the filters used in the process known as deep bed filtration. As a suspension flows through a filter composed of granular or fibrous materials, fines or colloidal particles penetrate the filter and deposit at various depths [32]. As a result, the fluid suspension is cleaner when it exits the filter (i.e., it exits the filter with many fewer solid particles than it originally had when it entered the filter).Theoretical models to study transport in porous media can be classified as either macro-scale [5,15,22,23,24,26,32] or pore-scale [9,19,28] models. Within the latter group, the class of network models, in which the pore space is modeled as a network of channels, is very popular. Network models provide flexibility in modeling different geometries of pore space while keeping the computational cost low. Our work belongs to this class of models.Network models to study transport in porous media were introduced by Fatt in 1956 [10,11,12]. Donaldson, Baker, and Carrol in 1977 [8] were the first to use networks to study particle transport within porous media. The clogging of particles has been studied in networks with different geometries including bundles of parallel tubes [8], square networks [14,16,21], triangular networks [3,25], cubic networks

Abstract. Fluid in porous media flows through tortuous paths. When the fluid velocities are large enough, this tortuosity and inertial effects cause suspended particles to collide with pore walls. After each collision, a particle loses momentum and needs to be accelerated again by hydrodynamic forces. As a result, the average velocity of suspended particles may be smaller than that of the fluid. In addition, if the fluid velocity field is not macroscopically homogeneous, the retardation of the particles with respect to the fluid leads to an increase of the concentration of particles in certain regions which may eventually result in the clogging of the porous medium. This effect is of importance in flows near wells where the flow has circular symmetry and thus it is not macroscopically homogeneous. In this paper, we study the physical effect described above. We first develop a mathematical model that takes into account tortuosity and inertial effects at the pore scale. This model provides us with the average particle velocities as a function of the fluid velocity. We use this function in a second model, a macroscopic scale model, to study the evolution of concentration of particles and the development of clogging when the flow has circular symmetry.