We are interested in a viscous two-phase gas-liquid mixture model relevantfor modeling of well control operations within the petroleum industry. We focus on asimplified mixture model and provide an existence result within an appropriate class ofweak solutions. We demonstrate that upper and lower limits can be obtained for the gasand liquid masses which ensure that transition to single-phase regions do not occur. This isused together with appropriate a prior estimates to obtain convergence to a weak solutionfor a sequence of approximate solutions corresponding to mollified initial data. Moreover,by imposing an additional regularity condition on the initial masses, a uniqueness resultis obtained. The framework herein seems useful for further investigations of more realisticversions of the gas-liquid model that take into account different flow regimes.