Recently, fully variational discretizations of the dynamic poroelasticity system, rewritten as a first-order system in space and time, have been proposed and studied [M. Bause, S. Franz, M. Anselmann, Structure preserving discontinuous Galerkin approximation of a hyperbolic-parabolic system, Electron. Trans. Numer. Anal., 63 (2025), pp. 1--32]. The mathematical structure of the continuous system is preserved on the discrete level, accurate stress and flux approximation as part of the formulation itself. For solving the algebraic systems, applying GMRES iterations with V-cycle geometric multigrid preconditioning has proved to be robust and efficient [M. Anselmann, M. Bause, N. Margenberg, P. Shamko, An energy-efficient GMRES--Multigrid solver for space-time finite element computation of dynamic poroelasticity, Comput. Mech., 74 (2024), pp.\ 889--912]. This is illustrated here by numerical experiments. In applications that are of interest in practice, the definition of the properties of porous media in space can only be made by the concept of random functions. For this reason, stochastic perturbations need to be integrated into mathematical modeling and numerical simulation of poroelasticity. In this contribution, we enlighten the potential of applying stochastic Galerkin methods for numerical modeling in complex random media. This is done for the prototype problem of nonstationary linear diffusion. Instead of using Monte-Carlo simulation techniques and generating parameter inputs randomly from a probability distribution over the domain, performing deterministic computations of the outputs and aggregating the results, the variational problem is augmented by further dimensions linked to the stochastic variables. Appreciable advantable of this approach is that the holistic variational setting is preserved. Thereby, tensor product finite element discretization techniques, enabling matrix-free implementations with tailored solver for efficient high performance computing, become feasible [N. Margenberg, M. Bause, P. Munch, An hp multigrid approach for tensor-product space-time finite element discretizations of the Stokes equations, SIAM J. Sci. Comput., 47 (2025), pp.~1503--1529]. The stochastic Galerkin approach is introduced, and preliminary numerical results are presented.