Irreversible, dissipative processes can be often naturally modeled as gradient flows; under certain assumptions, flow in deformable porous media is such a process. In this talk, we formulate the problem of linear poro-(visco-)elasticity as generalized gradient flow. By exploiting this structure, the analysis of well-posedness and construction of numerical solvers can be performed in a rather straight-forward and abstract manner. For this, we apply results from gradient flow theory and convex optimization. We show that the standard iterative operator splitting schemes for linear poro-elasticity are equivalent with alternating minimization applied to quadratic optimization problems. Based on this observation, acceleration can be performed cheaply using a line search strategy. Using the same ideas, we derive new splitting schemes for poro-visco-elasticity with guaranteed convergence. Numerical examples are presented to verify the theory.