The applications of elastic membrane range from determining the mechanical properties of biological cells by indentation tests to predicting the deformed shape of a large commercial tent structure. In this work, direct membrane theory and a particular Varga strain energy function are used to model the indentation and puncturing of an isotropic spherical elastic membrane containing a fluid with a rigid indenter. The balance laws are applied to obtain the governing differential equations and numerical shooting method is used to solve them. Furthermore, a global mode of failure is established by computing the energy stored at the punctured membrane and this value determines a critical value for the energy of the membrane beyond which the punctured state of the membrane is energetically preferred. An additional mode of failure is identified in which the membrane loses local convexity requirements and it corresponds to the local loss of elastic behavior of the membrane.