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AbstractFiltration in general, and the dead end depth filtration of solid particles out of fluid in particular, is intrinsic multiscale problem. The deposition (capturing of particles) essentially depends on local velocity, on microgeometry (pore scale geometry) of the filtering medium and on the diameter distribution of the particles. The deposited (captured) particles change the microstructure of the porous media what leads to change of permeability. The changed permeability directly influences the velocity field and pressure distribution inside the filter element. To close the loop, we mention that the velocity influences the transport and deposition of particles. In certain cases one can evaluate the filtration efficiency considering only microscale or only macroscale models, but in general an accurate prediction of the filtration efficiency requires multiscale models and algorithms. This paper discusses the single scale and the multiscale models, and presents a fractional time step discretization algorithm for the multiscale problem. The velocity within the filter element is computed at macroscale, and is used as input for the solution of microscale problems at selected locations of the porous medium. The microscale problem is solved with respect to transport and capturing of individual particles, and its solution is postprocessed to provide permeability values for macroscale computations. Results from computational experiments with an oil filter are presented and discussed.