Rapid Inference of Reservoir Permeability From Inversion of Travel Time Data Under a Fast Marching Method

The Fast Marching Method (FMM) is a highly efficient numerical algorithm frequently used to solve the Eikonal equation to obtain the travel time from the source point to spatial locations, which can generate a geometric description of monotonically advancing front in anisotropic and heterogeneous media. In modeling fluid flow in subsurface heterogeneous porous media, application of the FMM makes the characterization of pressure front propagation quite straightforward using the diffusive time of flight (DTOF) as the Eikonal solution from an asymptotic approximation to the diffusivity equation. For the infinite-acting flow that occurs in smoothly varying heterogeneous media, travel time of 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 of investigation (ROI). Based on the ROI definition, the travel time to a given location in space can be determined from the maximum magnitude of partial derivative of pressure to time. Treating travel time computed at the observation well as the objective function, we propose a FMM based deep learning (DL) framework, namely the Inversion Neural Network (INN), to inversely estimate heterogeneous reservoir permeability fields through training the deep neural network (DNN) with the travel time data directly generated from the FMM. A convolutional neural network (CNN) is adopted to establish the mapping between the heterogeneous permeability field and the sparse observational data. Because of the quasi-linear relationship between the travel time and reservoir properties, CNN inspired by FMM is able to provide a rapid inverse estimate of heterogeneous reservoir properties that show sufficient accuracy compared to the true reference model with a limited number of observation wells. Inverse modeling results of the permeability fields are validated by the asymptotic pressure approximation through history matching of the reservoir models with the multi-well pressure transient data.