We consider an initial-boundary value problem for a system of nonlinear partial differential equations that describes two-phase, two-component fluid flow in porous media. The system equations are obtained from the mass conservation law written for each component. For main unknows we select persistent variables, capable of modelling the flow in both the two-phase flow and the one-phase flow regions. We use an artificial persistent variable called global pressure to rewrite the system in terms of global pressure and gas phase pseudo-pressure. As the obtained system contains degeneracy, we regularize it by a small parameter $\eta$. Furthermore, we apply time discretization, and introduce another persistent variable, called capillary pseudo-pressure. The global pressure partially decouples equations and we are able to obtain the existence of solution at discrete time level. By passing to the limit in the time discretization parameter and afterwards in regularization parameter $\eta$, we prove the existence of weak solutions of the introduced initial-boundary value problem.