  Simulation of multiphase flow in porous media is crucial for the effective
management of subsurface energy and environment related activities. The
numerical simulators used for modeling such processes rely on spatial and
temporal discretization of the governing partial-differential equations (PDEs)
into algebraic systems via numerical methods. These simulators usually require
dedicated software development and maintenance, and suffer low efficiency from
a runtime and memory standpoint. Therefore, developing cost-effective,
data-driven models can become a practical choice since deep learning approaches
are considered to be universal approximations. In this paper, we describe a
gradient-based deep neural network (GDNN) constrained by the physics related to
multiphase flow in porous media. We tackle the nonlinearity of flow in porous
media induced by rock heterogeneity, fluid properties and fluid-rock
interactions by decomposing the nonlinear PDEs into a dictionary of elementary
differential operators. We use a combination of operators to handle rock
spatial heterogeneity and fluid flow by advection. Since the augmented
differential operators are inherently related to the physics of fluid flow, we
treat them as first principles prior knowledge to regularize the GDNN training.
We use the example of pressure management at geologic CO2 storage sites, where
CO2 is injected in saline aquifers and brine is produced, and apply GDNN to
construct a predictive model that is trained from physics-based simulation data
and emulates the physics process. We demonstrate that GDNN can effectively
predict the nonlinear patterns of subsurface responses including the
temporal-spatial evolution of the pressure and saturation plumes. GDNN has
great potential to tackle challenging problems that are governed by highly
nonlinear physics and enables development of data-driven models with higher
fidelity.
