This dissertation addresses well-posedness and approximation problems for coupled parabolic-elliptic systems with applications to geomechanics. The work is motivated by problems of borehole stability in porous formations that involve the modeling of fully coupled thermal, chemical, hydraulic, and mechanical processes. Sufficient conditions for well-posedness of coupled parabolic-elliptic initial-boundary value problems for fully coupled thermo-chemo-poroelastic (TCPu) models are established. Fourier-finite-element methods for solving fully coupled two- and three-dimensional radially non-symmetric TCPu problems are developed. The obtained results are fundamental to posing optimal control problems in which the temperature and pressure on the borehole boundary are considered as control parameters used to achieve desired stresses in the neighborhood of the borehole.