This contribution focuses on the acceleration of sampling in Bayesian inversion using adaptively constructed surrogate models. The posterior distribution is analyzed using samples generated by Markov chain Monte Carlo (MCMC) methods. We use a sampling procedure based on the delayed-acceptance Metropolis-Hastings (DAMH) algorithm, which works with both the true posterior and its approximation. Like the basic Metropolis-Hastings (MH) algorithm, DAMH provides samples from the true posterior. The approximation serves only to accelerate the process. Unlike the MH algorithm, the forward mathematical model is not evaluated for every proposed sample. This can lead to a significant increase in efficiency; the more accurate the posterior approximation, the fewer evaluations are required. The sampling framework has been implemented as a Python library. The posterior approximation is constructed using a surrogate of the forward model. Surrogate models can be constructed using various techniques, such as interpolation with radial basis functions, polynomial approximation, or neural networks. The sampling framework is applied to a geotechnical inverse problem.