The aim of this work is to analyze a convolution quadrature boundary element approach to simulate wave propagation in porous media. In Laplace domain the model results in an elliptic second order partial differential equation. First, boundary value problems of interest are described and equivalent boundary integral formulations are derived. Unique solvability of all discussed boundary value problems and boundary integral equations is discussed, first in Laplace domain and fi nally also in time domain. A Galerkin discretization in space and a convolution quadrature discretization in time is applied. Unique solvability of the discrete systems and convergence of the approximate solutions are discussed. Finally, the theoretical results are confirmed by numerical experiments.