  We present a new hybrid physics-based machine-learning approach to reservoir
modeling. The methodology relies on a series of deep adversarial neural network
architecture with physics-based regularization. The network is used to simulate
the dynamic behavior of physical quantities (i.e. saturation) subject to a set
of governing laws (e.g. mass conservation) and corresponding boundary and
initial conditions. A residual equation is formed from the governing
partial-differential equation and used as part of the training. Derivatives of
the estimated physical quantities are computed using automatic differentiation
algorithms. This allows the model to avoid overfitting, by reducing the
variance and permits extrapolation beyond the range of the training data
including uncertainty implicitely derived from the distribution output of the
generative adversarial networks. The approach is used to simulate a 2 phase
immiscible transport problem (Buckley Leverett). From a very limited dataset,
the model learns the parameters of the governing equation and is able to
provide an accurate physical solution, both in terms of shock and rarefaction.
We demonstrate how this method can be applied in the context of a forward
simulation for continuous problems. The use of these models for the inverse
problem is also presented, where the model simultaneously learns the physical
laws and determines key uncertainty subsurface parameters. The proposed
methodology is a simple and elegant way to instill physical knowledge to
machine-learning algorithms. This alleviates the two most significant
shortcomings of machine-learning algorithms: the requirement for large datasets
and the reliability of extrapolation. The principles presented in this paper
can be generalized in innumerable ways in the future and should lead to a new
class of algorithms to solve both forward and inverse physical problems.
