Understanding of fluid behavior in fluid-saturated porous media has been a challenging research problem tackled through theoretical, experimental, and numerical approaches over many decades. Modeling such multiphysics phenomena, in both small and large deformation regimes, requires careful consideration of numerical accuracy and stability to achieve reliable results. This study employs the Material Point Method (MPM), a hybrid Eulerian-Lagrangian approach that can simulate large deformation problems involving history-dependent materials. The dissertation's contributions can be divided into two main application areas: (i) deep geomechanical well analysis under fluid injection and extraction, and (ii) surficial geotechnical structures involving the nonlinear interaction of free-surface flow and deformable porous media.

First, a new approach called CutMesh MPM is introduced for small-deformation solid mechanics. This method improves standard MPM discretization by re-evaluating quadrature weights and incorporating explicit boundary discretization to better track evolving surface geometry and apply boundary conditions. Algorithms for boundary detachment, detection, and reconstruction are introduced to track the evolution of wellbore geometry due to material detachment. The efficacy of the approach is validated through numerical studies of dry boreholes under anisotropic loading. Building on this, the study extends the CutMesh MPM framework to hydro-mechanical analysis of saturated boreholes. The coupled formulation integrates a two-phase single-point MPM formulation, assuming a fractional-step method, for accurate analysis of fluid extraction and injection in poroelastic and poro-elasto-plastic materials.

The second part of the dissertation focuses on modeling free-surface fluid dynamics, seepage, and interactions with porous solids. A mixed MPM formulation is introduced to address challenges related to the numerical instability of incompressible fluid flow. Several stabilization strategies are proposed and utilized to enhance the simulation's robustness and accuracy. This formulation is verified and validated through extensive numerical tests on complex fluid dynamics problems. Additionally, a unified approach integrating the Navier-Stokes and Darcy-Brinkman-Forchheimer equations is proposed to model flows in both non-porous and porous domains. This method uses the blurred interface technique to improve numerical stability at the interface, as demonstrated in a range of 1D, 2D, and 3D benchmarks. Finally, the mixed MPM is extended to account for strongly coupled hydro-mechanical problems through a three-field formulation involving solid displacement, fluid displacement, and pressure. The proposed approach features stabilized discretization methods and consistent derivations, with preliminary results in three-dimensional consolidation problems demonstrating its potential.