A kind of compressible miscible displacement problems which include
molecular diffusion and dispersion in porous media are investigated.
The mixed finite element method is applied to the flow equation, and the
transport one is solved by the symmetric interior penalty
discontinuous Galerkin method. Based on a duality argument,
employing projection estimates and approximation properties, a
posteriori residual-type $hp$ error estimates for the coupled system
are presented, which is often used for guiding adaptivity. Comparing
with the error analysis carried out by Yang (Int. J. Numer. Meth.
Fluids, 65(7) (2011), pp. 781-797), the current work is more
complicated and challenging.