  In this work, the space-time MORe DWR (Model Order Reduction with
Dual-Weighted Residual error estimates) framework is extended and further
developed for single-phase flow problems in porous media. Specifically, our
problem statement is the Biot system which consists of vector-valued
displacements (geomechanics) coupled to a Darcy flow pressure equation. The
MORe DWR method introduces a goal-oriented adaptive incremental proper
orthogonal decomposition (POD) based-reduced-order model (ROM). The error in
the reduced goal functional is estimated during the simulation, and the POD
basis is enriched on-the-fly if the estimate exceeds a given threshold. This
results in a reduction of the total number of full-order-model solves for the
simulation of the porous medium, a robust estimation of the quantity of
interest and well-suited reduced bases for the problem at hand. We apply a
space-time Galerkin discretization with Taylor-Hood elements in space and a
discontinuous Galerkin method with piecewise constant functions in time. The
latter is well-known to be similar to the backward Euler scheme. We demonstrate
the efficiency of our method on the well-known two-dimensional Mandel benchmark
and a three-dimensional footing problem.
