For strongly non-linear and high-dimensional inverse problems, Markov chain Monte Carlo (MCMC) methods may fail to properly explore the posterior probability density function (PDF) given a realistic computational budget and are generally poorly amenable to parallelization. Particle methods approximate the posterior PDF using the states and weights of a population of evolving particles and they are very well suited to parallelization. We focus on adaptive sequential Monte Carlo (ASMC), an extension of annealed importance sampling (AIS). In AIS and ASMC, importance sampling is performed over a sequence of intermediate distributions, known as power posteriors, linking the prior to the posterior PDF. The AIS and ASMC algorithms also provide estimates of the evidence (marginal likelihood) as needed for Bayesian model selection, at basically no additional cost. ASMC performs better than AIS as it adaptively tunes the tempering schedule and performs resampling of particles when the variance of the particle weights becomes too large. We consider a challenging synthetic groundwater transport inverse problem with a categorical channelized 2D hydraulic conductivity field defined such that the posterior facies distribution includes two distinct modes. The model proposals are obtained by iteratively re-simulating a fraction of the current model using conditional multiple-point statistics (MPS) simulations. We examine how ASMC explores the posterior PDF and compare with results obtained with parallel tempering (PT), a state-of-the-art MCMC inversion approach that runs multiple interacting chains targeting different power posteriors. For a similar computational budget, ASMC outperforms PT as the ASMC-derived models fit the data better and recover the reference likelihood. Moreover, we show that ASMC partly retrieves both posterior modes, while none of them is recovered by PT. Lastly, we demonstrate how the power posteriors obtained by ASMC can be used to assess the influence of the assumed data errors on the posterior means and variances, as well as on the evidence. We suggest that ASMC can advantageously replace MCMC for solving many challenging inverse problems arising in the field of water resources.