  The calibration of a reservoir model with observed transient data of fluid
pressures and rates is a key task in obtaining a predictive model of the flow
and transport behaviour of the earth's subsurface. The model calibration task,
commonly referred to as "history matching", can be formalised as an ill-posed
inverse problem where we aim to find the underlying spatial distribution of
petrophysical properties that explain the observed dynamic data. We use a
generative adversarial network pretrained on geostatistical object-based models
to represent the distribution of rock properties for a synthetic model of a
hydrocarbon reservoir. The dynamic behaviour of the reservoir fluids is
modelled using a transient two-phase incompressible Darcy formulation. We
invert for the underlying reservoir properties by first modeling property
distributions using the pre-trained generative model then using the adjoint
equations of the forward problem to perform gradient descent on the latent
variables that control the output of the generative model. In addition to the
dynamic observation data, we include well rock-type constraints by introducing
an additional objective function. Our contribution shows that for a synthetic
test case, we are able to obtain solutions to the inverse problem by optimising
in the latent variable space of a deep generative model, given a set of
transient observations of a non-linear forward problem.
