In mathematical physics the problems of invariance with respect to the choice of reference systems are very important and interesting. The engineering assets of water networks are an example where problems of invariance can be encountered. In this paper we present a new extension, or variant, \(I_r^{(3)}\) of a well known resilience index for water distribution networks, originally introduced by E. Todini in 2000. In particular, we will prove that \(I_r^{(3)}\) is not affected by problems of invariance with respect to the reference frame even in the case of pressure deficits of a part of the nodes of the network or other conditions of any kind. More precisely we can say that the index \(I_r^{(3)}\) is intrinsically invariant.