**9.1 Fast marching method (FMM)**

 Classical reservoir simulation using finite difference and finite volume have been widely used to simulate fluid flow in conventional oil and gas reservoirs. With the advent of unconventional reservoirs, these widely used classical reservoir simulation tools have been adapted to the particular nature of permeability generated through hydraulic fracturing. Current state of the art and the challenges associated with unconventional reservoirs are discussed in other books [29]. In this chapter we address the issue of computation time and the need for unconventional reservoir simulators to provide an estimate of the EUR and pressure depletion in the fastest possible way to enable the engineers and geoscientist to compare multiple development scenarios very quickly. In other words, given the nature of the unconventional process and its fast pace, how we can trade a reduction in accuracy for a much faster reservoir simulation?

Multiple efforts have been made to reduce computation time in reservoir simulators by replacing the flow equations with a proxy model based on neural networks [30] or response surfaces and experimental design [31]. Other efforts include the use of fast front-tracking techniques using streamlines [32] and, more recently, using a fast marching method or FMM [33]. We use the FMM in our approach of modeling unconventional reservoirs for the 3D estimation of pressure depletion.

 The FMM is a front tracking algorithm. It has been applied to wave propagation, and medical imaging [34] problems. For the subsurface porous media flow, the pressure diffusivity equation can be simplified to an Eikonal equation and solved using the FMM to obtain the diffusive time of flight contours which is a proxy for the pressure depletion time [35]. The simulation is very fast compared to a finite difference simulator thus its valuable application to the fast-paced world of unconventional reservoirs. This speed is gained through a loss of accuracy which we could estimate by comparing the resulting pressure to those derived in a finite difference simulator.
