  This paper presents the unsteady Darcy's equations coupled with two nonlinear
reaction-diffusion equations, namely this system describes the mass
concentration and heat transfer in porous media. The existence and uniqueness
of the solution are established for both the variational formulation problem
and for its discrete one obtained using spectral discretization. Optimal a
priori estimates are given using the Brezzi-Rappaz-Raviart theorem. We conclude
by some numerical tests which are in agreement with our theoretical results.
