Hydrogeological field studies rely often on a single conceptual representation of the subsurface. This is problematic since the impact of a poorly chosen conceptual model on predictions might be significantly larger than the one caused by parameter uncertainty. Furthermore, conceptual models often need to incorporate geological concepts and patterns in order to provide meaningful uncertainty quantification and predictions. Consequently, several geologically realistic conceptual models should ideally be considered and evaluated in terms of their relative merits. Here, we propose a full Bayesian methodology based on Markov chain Monte Carlo to enable model selection among 2‐D conceptual models that are sampled using training images and concepts from multiple‐point statistics. More precisely, power posteriors for the different conceptual subsurface models are sampled using sequential geostatistical resampling and Graph Cuts. To demonstrate the methodology, we compare and rank five alternative conceptual geological models that have been proposed in the literature to describe aquifer heterogeneity at the MAcroDispersion Experiment site in Mississippi, USA. We consider a small‐scale tracer test for which the spatial distribution of hydraulic conductivity impacts multilevel solute concentration data observed along a 2‐D transect. The thermodynamic integration and the stepping‐stone sampling methods were used to compute the evidence and associated Bayes factors using the computed power posteriors. We find that both methods are compatible with multiple‐point statistics‐based inversions and provide a consistent ranking of the competing conceptual models considered.