We consider the mixed covolume method combining with the expanded mixed element for a system of first-order partial differential equations resulting from the mixed formulation of a general self-adjoint elliptic problem with a full diffusion tensor. The system can be used to model the transport of a contaminant carried by a flow in porous media. We use the lowest order Raviart-Thomas mixed element space. We show the first-order error estimate for the approximate solution in L2 norm. We show the superconvergence both for pressure and velocity in certain discrete norms. We also get a finite difference scheme by using proper approximate integration formulas. Finally we give some numerical examples.