The problem of formation damage, i.e., permeability reduction due to injection of particulates, is a matter of interest in several engineering fields. In the previous attempts to model the external cake formation, cake thickness has been considered to be only dependent on time; even though, in practical applications the dependency of the cake profile on space can be important. In this paper a novel model has been developed to describe the steady state external filter cake thickness profile along the well-bore. A set of equations are derived from the force balance for a deposited particle on the cake surface and the volume conservation of the fluid in the well-bore. These equations are combined with the Darcy's law in radial geometry and the equation of flow in the well-bore, and solved numerically to obtain the cake thickness and fluid velocity profiles along the well-bore.
Many oil and gas reservoirs are connected to water source connected to their hydrocarbon reserves. Therefore, the produced fluids are not exclusively hydrocarbons but contain water, often in very large amounts of water Worldwide 75% of the production is water  and in some regions this fraction may reach to 98% . Produced water (PW), depending on the geological formation and lifetime of the reservoir, contains organic and inorganic materials including solid particles and oil droplets. Disposal of the PW is a challenge for petroleum industry due to the high costs of the transportation and filtration facilities and the increasingly stricter environmental regulation.
One economically and environmentally friendly method is to re-inject the produced water into the formation. By re-injecting the produced water reservoir pressure can be maintained simultaneously. Nevertheless a rapid injectivity decline is usually observed in the PW injection processes due to the presence of impurities, usually solid particles and oil droplets. The injectivity decline is caused by the deposition of the particles inside porous media (internal filtration)  or accumulation of the particles on the surface (external filtration) . In the latter case, particles form a cake (porous medium), which is orders of magnitude less permeable than the reservoir.
Filtration of the particles in the borehole can occur under either static (dead-end) or dynamic (crossflow) flow conditions. Crossflow filtration refers to a pressure driven separation process in which the permeate flow is perpendicular to the feed flow. In the crossflow filtration, the thickness of the fiklter-cake is limited by the shear forces due to the flow of the produced water containg particles parallel to the borehole surface. A similar phenomenon happens when drilling mud flows across the borehole surface . The drag force caused by the flux of permeate pushes the suspended particles towards the cake.
Many authors have investigated the problem of crossflow filtration [6–17]. It has been suggested in the literature that temperature, pressure, shear rate (i.e., flow rate) and the permeability of the formation influences the filtration process. However, the effect of individual parameters has remained unclear .
In petroleum engineering transport of fluid containing particles in the subsurface is similar to crossflow filtration. For example, in the fractures, the transport of the particles in the injected fluid results in formation of the cake on the rock surface. Al-Abduwani et al.  developed a model to obtain the filter cake profiles in an experimental set-up. Their model can be applied for the formation of cake in the fractures (linear cases). Another example is the build-up of mud cake during the water injection or drilling of the well around the borehole. In this paper we adapt the model described in ref.  to obtain the filter cake and velocity profiles in a radial geometry.