  Machine learning-based data-driven modeling can allow computationally
efficient time-dependent solutions of PDEs, such as those that describe
subsurface multiphysical problems. In this work, our previous approach of
conditional generative adversarial networks (cGAN) developed for the solution
of steady-state problems involving highly heterogeneous material properties is
extended to time-dependent problems by adopting the concept of continuous cGAN
(CcGAN). The CcGAN that can condition continuous variables is developed to
incorporate the time domain through either element-wise addition or conditional
batch normalization. We note that this approach can accommodate other
continuous variables (e.g., Young's modulus) similar to the time domain, which
makes this framework highly flexible and extendable. Moreover, this framework
can handle training data that contain different timestamps and then predict
timestamps that do not exist in the training data. As a numerical example, the
transient response of the coupled poroelastic process is studied in two
different permeability fields: Zinn \& Harvey transformation and a bimodal
transformation. The proposed CcGAN uses heterogeneous permeability fields as
input parameters while pressure and displacement fields over time are model
output. Our results show that the model provides sufficient accuracy with
computational speed-up. This robust framework will enable us to perform
real-time reservoir management and robust uncertainty quantification in
realistic problems.
