In this talk we present ecient numerical schemes for linear and nonlinear Biot models [1, 2]. Nonlinear Lame coecients and/or uid compressibility or large deformations are considered. Furthermore, an erosion problem with a moving boundary is presented [3]. We use the L-scheme, see e.g. [6] or the Newton method for linearization, either monolithically or combined with a xed stress type splitting [5, 4]. Additionally, the optimization of the stabilization parameter in the xed-stress scheme will be discussed [7]. Bibliography [1] M. Borregales, F.A. Radu, K. Kumar, J.M. Nordbotten, Robust iterative schemes for nonlinear poromechanics, Comput. Geosci. 22 (2018): 1021-1038. [2] M. Borregales, F.A. Radu, K. Kumar, J.M. Nordbotten, F.A. Radu, Iterative solvers for Biot model under small and large deformations, Comput. Geosci. (2020). [3] D. Cerroni, F.A. Radu, P. Zunino, Numerical solvers for a poromechanics problem with a moving boundary, Mathematics in Engineering 1 (2019): 824-848. [4] J. Both, M. Borregales, F.A. Radu, K. Kumar, J.M. Nordbotten, Robust xed stress splitting for Biot's equations in heterogeneous media, Applied Mathematics Letters 68 (2017): 101-108. [5] J. Kim, H. Tchelepi, R. Juanes, Stability and convergence of sequential methods for coupledow and geomechanics: Fixed-stress and xed-strain splits, CMAME 200 (2011): 1591-1606. [6] F. List, F.A. Radu, A study on iterative methods for Richards' equation, Comput. Geosci. 20 (2016): 341-353. [7] E. Storvik, J. Both, K. Kumar, J.M. Nordbotten, F.A. Radu, On the optimization of the xed-stree splitting for Biot's equations, IJNME 120(2019): 179{194.