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Time discretization along with space discretization is important in the numerical simulation of subsurface flow applications for long run. In this paper, we derive theoretical convergence error estimates in discrete-time setting for transient problems with the Dirichlet boundary condition. Enhanced Velocity Mixed FEM as domain decomposition method is used in the space discretization and the backward Euler method and the Crank–Nicolson method are considered in the discrete-time setting. Enhanced Velocity scheme was used in the adaptive mesh refinement dealing with heterogeneous porous media [1,2] for single phase flow and transport and demonstrated as mass conservative and efficient method. Numerical tests validating the backward Euler theory are presented. These error estimates are useful in the determining of time step size and the space discretization size. [ABSTRACT FROM AUTHOR]

We propose an adaptive multiscale method to improve the efficiency and the accuracy of numerical computations by combining numerical homogenization and domain decomposition for modeling flow and transport. Our approach focuses on minimizing the use of fine scale properties associated with advection and diffusion/dispersion. Here a fine scale flow and transport problem is solved in subdomains defined by a transient region where spatial changes in transported species concentrations are large while a coarse scale problem is solved in the remaining subdomains. Away from the transient region, effective macroscopic properties are obtained using local numerical homogenization. An Enhanced Velocity Mixed Finite Element Method (EVMFEM) as a domain decomposition scheme is used to couple these coarse and fine subdomains [1]. Specifically, homogenization is employed here only when coarse and fine scale problems can be decoupled to extract temporal invariants in the form of effective parameters. In this paper, a number of numerical tests are presented for demonstrating the capabilities of this adaptive numerical homogenization approach in upscaling flow and transport in heterogeneous porous medium. • Numerical homogenization for effective property evaluation with mesh refinement. • Efficient and accurate representation of slightly compressible subsurface physics. • Verification, benchmarking and speedup reporting using datasets SPE 10 datasets. • Approach can handle many property distributions: Gaussian, layered, channelized. • Adaptivity criterion can be altered to obtain balance between speedup and accuracy. [ABSTRACT FROM AUTHOR]

Abstract A space–time domain decomposition approach is presented as a natural extension of the enhanced velocity mixed finite element (EVMFE), introduced by Wheeler et al. in (2002) [26] , for spatial domain decomposition. The proposed approach allows for different space–time discretizations on non-overlapping, subdomains by enforcing a mass continuity at non-matching interfaces to preserve local mass conservation inherent to the mixed finite element methods. To this effect, we consider three different model formulations: (1) a linear single phase flow problem, (2) a non-linear slightly compressible flow and tracer transport, and (3) a non-linear slightly compressible, multiphase flow and transport. We also present a numerical solution algorithm for the proposed domain decomposition approach where a monolithic (fully coupled in space and time) system is constructed that does not require subdomain iterations. This space–time EVMFE method accurately resolves advection–diffusion transport features, in a heterogeneous medium, while circumventing non-linear solver convergence issues associated with large time-step sizes for non-linear problems. Numerical results are presented for the aforementioned, three, model formulations to demonstrate the applicability of this approach to a general class of flow and transport problems in porous media. Highlights • Space time domain decomposition for spatial and temporal refinements in subdomains. • Fully implicit, monolithic, space–time solver. • Circumvent Newton convergence and small time-step size issues for non-linear PDEs. • Numerical results for scientific and practical problems of interest in porous medium. [ABSTRACT FROM AUTHOR]