Time-lapse Full Waveform Inversion for Subsurface Flow Problems with Intelligent Automatic Differentiation

We describe a novel framework for PDE (partial-differential-equation)-constrained full-waveform inversion (FWI) that estimates parameters of subsurface flow processes, such as rock permeability and porosity, using time-lapse observed data. The forward modeling couples flow physics, rock physics, and wave physics models. For the inverse modeling, we handle the back-propagation of gradients by an intelligent automatic differentiation strategy that offers three levels of user control: (1) At the wave physics level, we adopt the discrete adjoint method in order to use our existing high-performance FWI code; (2) at the rock physics level, we use built-in operators from the $\texttt{TensorFlow}$ backend; (3) at the flow physics level, we code customized PDE operators for the potential and nonlinear saturation equations, which highly resemble neural network layers. These three levels of gradient computation strike a good balance between computational efficiency and programming efficiency, and when chained together constitute the coupled inverse system. We use numerical experiments to demonstrate: (1) the three-level coupled inverse problem is superior in terms of accuracy to a traditional strategy; (2) it is able to simultaneously invert for parameters in empirical relationships such as the rock physics models; (3) and, the inverted model can be used for reservoir performance prediction and reservoir management/optimization purposes.