The contribution is motivated by the Bayesian approach to the solution of material identification problems which frequently appear in geo-engineering. We shall consider the cases with associated forward model describing flow in porous media with or without fractures as well as coupled hydro-mechanical processes. When assuming uncertainties in observed data, the use of the Bayesian inversion is natural. In comparison to deterministic methods, which lead only to a point estimate of the identified parameters, the Bayesian approach provides their probability distribution. The implementation of the Bayesian inversion is realized via Markov Chain Monte Carlo methods. The paper aims at the acceleration of the posterior sampling using a surrogate model that provides a polynomial approximation of the full forward model. The sampling procedure is based on the delayed acceptance Metropolis-Hastings (DAMH) algorithm. Therefore, for each proposed sample, the acceptance decision contains a preliminary step, which works only with an approximated posterior distribution constructed using the surrogate model. Furthermore, the approximated posterior distribution is being updated using new snapshots obtained during the sampling process. The posterior distribution updates are realized via updates of the surrogate model. The application of the described approach is shown through several model examples including flow in porous media with fractures and hydro-mechanical coupling.