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Integrated Asset Management and Optimization Workflows 217

of reservoir AHM, while Hajizadeh (2011) gives a very comprehensive
review of the evolution of history matching over the last 50years.

6.3.1 Model Parameterization and Dimensionality Reduction
Dynamic calibration of reservoir models is an ill-posed and potentially unsta-
ble inverse problem, where many probable solutions can satisfy the posed
objective function. This approach often requires a re-definition of subsur-
face spatial properties into parameter groups that provide a tool for problem
regularization and make the inversion problem computationally tractable.
Such techniques are commonly referred to as “parameterization techniques”
and in their applications to history matching and model calibration their
goal is primarily to replace the original set of unknown spatially discretized
reservoir properties with a reduced (lower-dimensional) number of para-
meters while retaining minimum possible loss of information density in rep-
resenting the most representative features of the reservoir model.
An example is a parameterization of the depositional system and under-
lying facies distribution to strategically group the most dominant flow units
while retaining the inherent spatial heterogeneity and continuity that drive
the reservoir connectivity. Bhark et al. (2011) provide an in-depth review of
the most prevalent model parameterization techniques, while Hutahaean
et al. (2015) and Al-Shamma et al. (2015) study their impact on the objective
choices with implications to simulation model history matching.
Zonation (and its adaptive variants) has traditionally been used in petro-
leum applications for adjusting reservoir model static properties like poros-
ity, permeability, and transmissibility. Herewith, however, we briefly
present the subspace and low-rank approximation methods that provide
the basis for multiscale parameterization based on linear transformation
for applications in AHM and optimization. In brief, the methods of linear
transformation map the spatial parameters (i.e., reservoir parameters, like
porosity or permeability, subject to parameterization) from the
“parameter domain” to the transformed, low-rank (i.e., low dimension)
“feature domain”, where parameter estimation and model updating can
be performed in a more efficient manner. Using an orthogonal transform,
the discrete spatial variable u is mapped to the transform domain as
(Bhark et al., 2011).

T
v ¼ Φ u , u ¼ Φv (6.7)