  Subsurface flow problems usually involve some degree of uncertainty.
Consequently, uncertainty quantification is commonly necessary for subsurface
flow prediction. In this work, we propose a methodology for efficient
uncertainty quantification for dynamic subsurface flow with a surrogate
constructed by the Theory-guided Neural Network (TgNN). The TgNN here is
specially designed for problems with stochastic parameters. In the TgNN,
stochastic parameters, time and location comprise the input of the neural
network, while the quantity of interest is the output. The neural network is
trained with available simulation data, while being simultaneously guided by
theory (e.g., the governing equation, boundary conditions, initial conditions,
etc.) of the underlying problem. The trained neural network can predict
solutions of subsurface flow problems with new stochastic parameters. With the
TgNN surrogate, the Monte Carlo (MC) method can be efficiently implemented for
uncertainty quantification. The proposed methodology is evaluated with
two-dimensional dynamic saturated flow problems in porous medium. Numerical
results show that the TgNN based surrogate can significantly improve the
efficiency of uncertainty quantification tasks compared with simulation based
implementation. Further investigations regarding stochastic fields with smaller
correlation length, larger variance, changing boundary values and
out-of-distribution variances are performed, and satisfactory results are
obtained.
