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In coupled flow and poromechanics phenomena representing hydrocarbon production or CO₂ sequestration in deep subsurface non-fractured reservoirs, the spatial domain in which fluid flow occurs is usually much smaller than the spatial domain over which significant deformation occurs. The vertical extent of the poromechanical domain can be two orders of magnitude more than the characteristic thickness of the flow domain (reservoir). The lateral extent of the poromechanical domain should also be allowed to be substantially larger than that of the flow domain to enable the imposition of far-field boundary conditions on the poromechanical domain. The typical approach is to either impose an overburden pressure directly on the reservoir thus treating it as a coupled problem domain or to model flow on a huge domain with zero permeability cells to mimic the no flow boundary condition on the interface of the reservoir and the surrounding rock. The former approach precludes a study of land subsidence or uplift and further does not mimic the true effect of the overburden on stress sensitive reservoirs whereas the latter approach has huge computational costs. The flow domain requires an areal resolution fine enough to be able to capture the underlying nonlinearities in the multiphase flow equations. If the same grid resolution is employed for the poromechanical domain, the simulator would crash for lack of memory and computing resource. With that in mind, it is imperative to establish a framework in which fluid flow is resolved on a finer grid and poromechanical deformation is resolved on a coarse grid. In addition, the geometry of the flow domain necessitates the use of non-nested grids which allows for freedom of choice of the poromechanical grid resolution. Furthermore, to achieve the goal of rendering realistic simulations of subsurface phenomena, we cannot ignore the heterogeneity in flow and poromechanical properties, as well as the lack in accuracy of the poromechanical calculations if the grid for the poromechanics domain is too coarse. This dissertation is a rendition of how we invoke concepts in computational geometry, parallel computing, applied mathematics and convex optimization in designing and implementing algorithms that tackle all the aforementioned challenges.