  This paper presents a non-intrusive subdomain POD-TPWL (SD POD-TPWL)
algorithm for reservoir data assimilation through integrating domain
decomposition (DD), radial basis function (RBF) interpolation and the
trajectory piecewise linearization (TPWL). It is an efficient approach for
model reduction and linearization of general non-linear time-dependent
dynamical systems without intruding the legacy source code. In the subdomain
POD-TPWL algorithm, firstly, a sequence of snapshots over the entire
computational domain are saved and then partitioned into subdomains. From the
local sequence of snapshots over each subdomain, a number of local basis
vectors is formed using POD, and then the RBF interpolation is used to estimate
the derivative matrices for each subdomain. Finally, those derivative matrices
are substituted into a POD-TPWL algorithm to form a reduced-order linear model
in each subdomain. This reduced-order linear model makes the implementation of
the adjoint easy and resulting in an efficient adjoint-based parameter
estimation procedure. The performance of the new adjoint-based parameter
estimation algorithm has been assessed through several synthetic cases.
Comparisons with the classic finite-difference based history matching show that
our proposed subdomain POD-TPWL approach is obtaining comparable results. The
number of full-order model simulations required is roughly 2-3 times the number
of uncertain parameters. Using different background parameter realizations, our
approach efficiently generates an ensemble of calibrated models without
additional full-order model simulations.
