  A posteriori error estimates are constructed for the three-field variational
formulation of the Biot problem involving the displacements, the total pressure
and the fluid pressure. The discretization under focus is the
H1({\Omega})-conforming Taylor-Hood finite element combination, consisting of
polynomial degrees k + 1 for the displacements and the fluid pressure and k for
the total pressure. An a posteriori error estimator is derived on the basis of
H(div)-conforming reconstructions of the stress and flux approximations. The
symmetry of the reconstructed stress is allowed to be satisfied only weakly.
The reconstructions can be performed locally on a set of vertex patches and
lead to a guaranteed upper bound for the error with a constant that depends
only on local constants associated with the patches and thus on the shape
regularity of the triangulation. Particular emphasis is given to nearly
incompressible materials and the error estimates hold uniformly in the
incompressible limit. Numerical results on the L-shaped domain confirm the
theory and the suitable use of the error estimator in adaptive strategies.
