The fast marching method (FMM) is a highly efficient numerical algorithm used to solve the Eikonal equation. It calculates traveltime from the source point to different spatial locations and provides a geometric description of the advancing front in anisotropic and heterogeneous media. As the Eikonal solution, the diffusive time of flight (DTOF) can be used to formulate an asymptotic approximation to the pressure diffusivity equation to describe transient flow behavior in subsurface porous media.

For the infinite-acting flow that occurs in porous media with smoothly varying heterogeneity, traveltime of the pressure front from the active production or injection well to the observation well can be directly estimated from the DTOF using the concept of radius (or depth) of investigation (ROI or DOI), which is defined as the moment when a maximum magnitude of the partial derivative of pressure to time occurs. Based on the ROI or DOI definition, we propose a deep neural network called the inversion neural network (INN) to inversely estimate heterogeneous reservoir permeability by inverting the traveltime data.

The INN is trained by traveltime data created for a large data set of distinct permeability fields from FMM simulations, which can be two orders of magnitude faster than conventional reservoir simulators. A convolutional neural network (CNN), the U-Net architecture, is incorporated into the INN, which establishes a nonlinear mapping between the heterogeneous permeability fields and the traveltime data collected at sparse observation wells. The loss function used for the INN is defined as the root mean square error (RMSE) between the logarithm of the predicted permeability and the logarithm of the true permeability.

The performance of the INN is tested on reservoir models with both smoothly varying heterogeneity and high-contrast media properties. For the 2D smoothly varying heterogeneous models with a grid size of 49×49, the permeability predicted by the INN has an average estimation error of 8.73% when a set of 7×7 uniformly distributed observation wells is used to collect “observational” traveltime data from the FMM simulation. For models with the same grid size and observation well density but with high-contrast media properties, the INN can still capture the general heterogeneity distribution, although with reduced prediction accuracy. Using a graphics processing unit (GPU) for training and prediction allows the entire inverse modeling process for a 2D 49×49 reservoir model to be completed within 7 minutes.