The problems encountered in modeling multistage hydraulic fracturing (MHF) in heterogeneous reservoirs are a widely discussed topic. Existing modeling tools, such as 3D simulation models, require a lot of time and resources. In order to reduce the computational resources and time spent on modeling complex problems, a method has been developed to reduce a three-dimensional problem to a one-dimensional one using the eikonal equation.The method of solving the diffusion equation using diffusive time of flight (DTOF) was developed and patented in the USA in 2015. According to the text of this patent, the main purpose of the method is to effectively select the design of MHF for a horizontal well (HW) in low-permeability and heterogeneous reservoirs. In practice, estimating drainage volume in heterogeneous, low-permeability reservoirs is a difficult task, which to date has been solved using 3D hydrodynamic simulation models (HDMs) due to the inability to use analytical methods.This paper discusses the application of the DTOF method for data retrieval and solution of the diffusion equation taking into account heterogeneous reservoir properties, as well as the parameterization problem using parameterization approaches. In addition, a solution algorithm based on the DTOF method and one of the parameterization approaches for automatic history matching to historical flow rates is constructed.
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