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Fully coupled flow-deformation analysis of deformable multiphase porous media saturated by several immiscible fluids has attracted the attention of researchers in widely different fields of engineering. This paper presents a new numerical tool to simulate the complicated process of two-phase fluid flow through deforming porous materials using a mesh-free technique, called element-free Galerkin (EFG) method. The numerical treatment of the governing partial differential equations involving the equilibrium and continuity equations of pore fluids is based on Galerkin’s weighted residual approach and employing the penalty method to introduce the essential boundary conditions into the weak forms....