  Linear poroelasticity models have a number of important applications in
biology and geophysics. In particular, Biot's consolidation model is a
well-known model that describes the coupled interaction between the linear
response of a porous elastic medium and a diffusive fluid flow within it,
assuming small deformations. Although deterministic linear poroelasticity
models and finite element methods for solving them numerically have been well
studied, there is little work to date on robust algorithms for solving
poroelasticity models with uncertain inputs and for performing uncertainty
quantification (UQ). The Biot model has a number of important physical
parameters and inputs whose precise values are often uncertain in real world
scenarios. In this work, we introduce and analyse the well-posedness of a new
five-field model with uncertain and spatially varying Young's modulus and
hydraulic conductivity field. By working with a properly weighted norm, we
establish that the weak solution is stable with respect to variations in key
physical parameters, including the Poisson ratio. We then introduce a novel
locking-free stochastic Galerkin mixed finite element method that is robust in
the incompressible limit. Armed with the `right' norm, we construct a
parameter-robust preconditioner for the associated discrete systems. Our new
method facilitates forward UQ, allowing efficient calculation of statistical
quantities of interest and is provably robust with respect to variations in the
Poisson ratio, the Biot--Willis constant and the storage coefficient, as well
as the discretization parameters.
