  Computer models are widely used across a range of scientific disciplines to
describe various complex physical systems, however to perform full uncertainty
quantification we often need to employ emulators. An emulator is a fast
statistical construct that mimics the slow to evaluate computer model, and
greatly aids the vastly more computationally intensive uncertainty
quantification calculations that an important scientific analysis often
requires. We examine the problem of emulating computer models that possess
multiple, partial discontinuities occurring at known non-linear location. We
introduce the TENSE framework, based on carefully designed correlation
structures that respect the discontinuities while enabling full exploitation of
any smoothness/continuity elsewhere. This leads to a single emulator object
that can be updated by all runs simultaneously, and also used for efficient
design. This approach avoids having to split the input space into multiple
subregions. We apply the TENSE framework to the TNO Challenge II, emulating the
OLYMPUS reservoir model, which possess multiple such discontinuities.
