In this paper, we are interested in the finite volume approximation of a system describing miscible displacement in porous media. This system is made of two coupled equations: an anisotropic diffusion equation on the pressure and a convection-diffusion-dispersion equation on the concentration of invading fluid. The anisotropic diffusion operators in both equations require a special care while discretizing by a finite volume method. We focus here on the numerical approximation by some Discrete Duality Finite Volume methods. After the presentation of the scheme, we establish some a priori estimates satisfied by the numerical solution and prove existence and uniqueness of the solution to the scheme. We show the efficiency of the schemes through numerical experiments.