Fluids with particles in suspension can be extracted through an orifice until particles bridge across the opening and clog it. We derive and study a mathematical model to predict the volume of fluid that can be extracted before clogging. Numerical simulations and closed-form upper and lower bound solutions show the dependency of the expected extracted volume on the volume fraction of particles in suspension and the relative particle-to-orifice size. These results are the first step toward a comprehensive understanding of the physical mechanisms leading to filter clogging with relevance to geomaterials, biomaterials, and industrial applications.