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In this paper, we consider the numerical solution of the poroelasticity problem with stochastic properties. We present a Two-stage Markov Chain Monte Carlo method for geomechanical subsidence. In this work, we study two techniques of preconditioning: (MS) multiscale method for model order reduction and (ML) machine learning technique. The purpose of preconditioning is the fast sampling, where a new proposal is first testes by a cheap multiscale solver or using fast prediction of the neural network and the full fine grid computations will be conducted only if the proposal passes the first step. To construct a reduced order model, we use the Generalized Multiscale Finite Element Method and present construction of the multiscale basis functions for pressure and displacements in stochastic fields. In order to construct a machine learning based preconditioning, we generate a dataset using a multiscale solver and use it to train neural networks. The Karhunen-Loeve expansion is used to represent the realization of the stochastic field. Numerical results are presented for two- and three-dimensional model examples.