We consider the motion of internal gravity waves at the interface between two immiscible inviscid fluids of different densities in the case where the top water surface of the upper layer is assumed to be flat. As in the case of gravity water waves, the basic equations have a variational structure with a Lagrangian in terms of the surface elevation of the interface and the velocity potentials of the two fluids. Kakinuma model is the Euler-Lagrange equation for an approximate Lagrangian, which is derived from the original Lagrangian by approximating the velocity potentials appropriately. We show basic structures of the Kakinuma model, especially, the linear dispersion relation, which implies that the Kakinuma model would be a good approximation to the original model in the shallow water regime. Although the initial value problem to the original model is ill-posed, the problem to the Kakinuma model turns out to be well-posed in an appropriate condition on the initial data. This talk is based on a joint work with Vincent Duchene (Universite de Rennes 1).