The new computational model for multiphase flow in deforming elastic porous media is proposed. The derivation of the model is based on the thermodynamically compatible hyperbolic systems of conservation laws theory. The flow of the mixture of compressible fluids in the elastic medium is supposed to be a continuum, in which the multiphase character of flow is taken into account. This phenomenological approach of continuum mechanics modelling allows us to formulate the system of governing equations in a divergent form, which is advantageous for the mathematical study of the different problems and for the development of advanced numerical methods. We present a thermodynamically compatible model for the flow of fluids mixture in elastic porous medium. The governing equations comprise balance laws for phase masses, total momentum and total energy supplemented by the equations for relative velocities in divergent form. The high accuracy Runge-Kutta-WENO numerical method for solving equations of the model is presened along the numerical test problem. The proposed model and developed numerical framework can be used in the wide range of oil recovery problems. Examples are: tracking oil/water interfaces in oil reservoirs, modeling of flows in the well surrounding formation