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Yield-stress materials or yield-stress fluids represent one of the most important categories of non-Newtonian fluids.Interaction of a yield-stress fluid with another fluid is of essential importance in fields like biotechnology, chemical engineering and petroleum industry, In order to investigate multiphase yield-stress fluid flow, we present an extended multiphase lattice Boltzmann model by applying the Herschel-Bulkley constitutive relationship to account for the variable viscosity.The lattice Boltzmann method (LBM) in this work is based on the free-energy theory, which is accurate and efficient to simulate fluids with high-density and high-viscosity contrasts.