  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.
