A multi-stage process for recovering acid gas from natural gas having high acid gas contents utilizes two or more membrane absorption contactors arranged in series. The first membrane absorption contactor uses a physical solvent to remove a high volume of acid gas transferred across a membrane, and to reduce the acid gas content in the natural gas to a lower level that can be managed using chemical solvents. The second and, if needed, subsequent membrane absorption contactors can use a chemical solvent to remove acid gas transferred across the respective membranes and reduce the acid gas content in the natural gas to very low levels, if needed, depending on product specifications.