  We present an adaptive space-time mesh refinement approach based a domain
decomposition approach (Singh and Wheeler, 2018) that allows different
time-step sizes and mesh refinements in different subdomains. Our numerical
experiments indicate that non-linear solvers fail to converge, to the desired
tolerance, due to large non-linear residuals in a smaller subdomain. We exploit
this feature to identify subdomains where smaller time-step sizes are necessary
while using large time-step sizes in the rest of the reservoir domain. The
three key components of our approach are: (1) a space-time, enhanced velocity,
domain decomposition approach that allows different mesh refinements and
time-step sizes in different subdomains while preserving local mass
conservation, (2) a residual based error estimator to identify or mark regions
(or subdomains) that pose non-linear convergence issues, and (3) a fully
coupled monolithic solver is also presented that solves the coarse and fine
subdomain problems, both in space and time, simultaneously. This solution
scheme is fully implicit and is therefore unconditionally stable. The proposed
space-time domain decomposition approach, with smaller time-step sizes in a
subdomain and large time-step sizes everywhere else, circumvents the non-linear
convergence issue without adding computational costs. Additionally, a
space-time monolithic solver renders a massively parallel, time concurrent
framework for solving flow and transport problems in subsurface porous media.
Since the proposed approach is similar to the widely used finite difference
scheme, it can be easily integrated in any existing legacy reservoir simulator.
