This article analyses the convergence of the Vertex Approximate Gradient (VAG) scheme recently introduced for the discretization of multiphase Darcy flows on general polyhedral meshes. The convergence of the scheme to a weak solution is shown in the particular case of an incompressible immiscible two phase Darcy flow model with capillary diffusion using a global pressure formulation. A remarkable property in practice is that the convergence is proven whatever the distribution of the volumes at the cell centres and at the vertices used in the control volume discretization of the saturation equation. The numerical experiments carried out for various families of 2D and 3D meshes confirm this result on a one dimensional Buckley Leverett solution.