This work explores the potential of the plurigaussian (PGS) method to create geostatistical models in which the facies connectivity as well as the facies proportion is an independent, predefined target parameter. The models considered are two- or three-dimensional and contain two facies. Connectivity is quantified by a parameter expressing how close the volume fraction of the foreground facies is to its percolation threshold (PT). A truncation scheme is developed in which each Gaussian Random Field (GRF) is truncated twice, with the central region defining the background facies and the other two regions defining the foreground facies of interest. In an n-dimensional PGS sampling scheme, the foreground facies is split into 2n mutually unconnected regions, and is macroscopically connected only if the proportion within the largest region exceeds PT. Altering the number of GRFs and the asymmetry of their truncations allows models to be created at specific facies proportions with different facies connectivity. Truncated Gaussian (TGS) models containing only one GRF are able to generate connected or unconnected 2D models at any facies proportions, but PGS models based on three or more GRFs are required in a 3D model. This work introduces the problem, discusses the analytical formulation of the approach developed, shows models and results that test it, and describes the modelling steps devised to create the models and constrain them to hard, observational well data.