In this paper, a series of multimaterial benchmark problems in saturated and partially saturated two-phase and three-phase deforming porous media are addressed. To solve the process of fluid flow in partially saturated porous media, a fully coupled three-phase formulation is developed on the basis of available experimental relations for updating saturation and permeabilities during the analysis. The well-known element free Galerkin mesh-free method is adopted. The partition of unity property of MLS shape functions allows for the field variables to be extrinsically enriched by appropriate functions that introduce existing discontinuities in the solution field. Enrichment of the main unknowns including solid displacement, water phase pressure, and gas phase pressure are accounted for, and a suitable enrichment strategy for different discontinuity types are discussed. In the case of weak discontinuity, the enrichment technique previously used by Krongauz and Belytschko [Int. J. Numer. Meth. Engng., 1998; 41:1215–1233] is selected. As these functions possess discontinuity in their first derivatives, they can be used for modeling material interfaces, generating only minor oscillations in derivative fields (strain and pressure gradients for multiphase porous media), as opposed to unenriched and constrained mesh-free methods. Different problems of multimaterial poro-elasticity including fully saturated, partially saturated one, and two-phase flows under the assumption of fully coupled extended formulation of Biot are examined. As a further development, problems involved with both material interface and impermeable discontinuities, where no fluid exchange is permitted across the discontinuity, are considered and numerically discussed. Copyright © 2014 John Wiley & Sons, Ltd.