Efficient algorithms for flow models coupled with geomechanics for porous media applications

The coupling between subsurface flow and reservoir geomechanics plays a critical role in obtaining accurate results for models involving reservoir deformation, surface subsidence, well stability, sand production, waste deposition, hydraulic fracturing, CO₂ sequestration, and hydrocarbon recovery. From a pure computational point of view, such a coupling can be quite a challenging and complicated task. This stems from the fact that the constitutive equations governing geomechanical deformations are different in nature from those governing porous media flow. The geomechanical effects account for the influence of deformations in the porous media caused due to the pore pressure and can be very important especially in the case of stress-sensitive and fractured reservoirs. Considering that fractures are very much prevalent in the porous media and they have strong influence on the flow profiles, it is important to study coupled geomechanics and flow problems in fractured reservoirs. In this work, we pursue three main objectives: first, to rigorously design and analyze iterative and explicit coupling algorithms for coupling flow and geomechanics in both poro-elasitc and fractured poro-elastic reservoirs. The analysis of iterative coupling schemes relies on studying the equations satisfied by the difference of iterates and using a Banach contraction argument to derive geometric convergence (Banach fixed-point contraction) results. The analysis of explicit coupling schemes result in analogous stability estimates. In this work, conformal Galerkin is used for mechanics, and a mixed formulation, including the Multipoint Flux Mixed Finite Element method as a special case, is used for the flow model. For fractured poro-elastic media, our iteratively coupled schemes are adaptations, due to the presence of fractures, of the classical fixed stress-splitting scheme, in which fractures are treated as possibly non-planar interfaces. The second main objective in this work is to exploit the different time scales of the mechanics and flow problems. Due to its physical nature, the geomechanics problem can cope with a coarser time step compared to the flow problem. This makes the multirate coupling scheme, the one in which the flow problem takes several (finer) time steps within the same coarse mechanics time step, a natural candidate in this setting. Inspired by that, we rigorously formulate and analyze convergence properties of both multirate iterative and explicit coupling schemes in both poro-elastic and fractured poro-elastic reservoirs. In addition, our theoretically derived Banach contraction estimates are validated against numerical simulations. The third objective in this work is to optimize the solution strategy of the nonlinear flow model in coupled flow and mechanics schemes. The global inexact Newton method, combined with the line search backtracking algorithm along with heuristic forcing functions, can be efficiently employed to reduce the number of flow linear iterations, and hence, the overall CPU run time. We first validate these computational savings for challenging two-phase benchmark problems including the full SPE10 model. Motivated by the obtained results, we incorporate this strategy as a nonlinear solver framework to solve the nonlinear flow problem in multirate iteratively coupled schemes. This leads to a scheme that reduces both the number of flow and mechanics linear iterations efficiently. All our numerical implementations in this work are built on top of our in-house reservoir simulator (IPARS).