  The existence of large-data weak entropy solutions to a nonisothermal
immiscible compressible two-phase unsaturated flow model in porous media is
proved. The model is thermodynamically consistent and includes temperature
gradients and cross-diffusion effects. Due to the fact that some terms from the
total energy balance are non-integrable in the classical weak sense, we
consider so-called variational entropy solutions. A priori estimates are
derived from the entropy balance and the total energy balance. The compactness
is achieved by using the Div-Curl lemma.
