We consider a Lagrangian algorithm for modeling the flow of hydraulic fracturing materials during a stimulation treatment of subterranean formation. The computational cost and accuracy associated with Lagrangian methods for particle transport simulation are directly related to the number of particles used for the representation of slurry components. This paper studies the problem of particle injection having a strong impact on the total particle count. We propose an adaptive algorithm analyzing the trajectories emanating from several points on the inflow perimeter. It allows one to achieve "almost uniform" distribution of particles in the flow domain after travelling a given distance (or time period). The capabilities of this algorithm are shown in numerical experiments, where we compare it with simple uniform particles seeding on inflow boundaries. It is seen that the uniform seeding leads to a poor quality of particles distribution on inflow segments where the direction of trajectories varies strongly. In addition, it results in excessive particles seeding in some other cases, which affects performance.
A numerical model of solids transport in a narrow channel is an essential part of any hydraulic fracturing simulator. Solids placement inside a fracture not only influences the post-fracturing production of hydrocarbons but also can greatly affect fracture propagation in the rock via different bridging scenarios and up to complete arrest of fracture propagation. Thus it is important to properly simulate solids transport in the fracture.
One of the ways to achieve proper simulation is to use a Lagrangian approach to solve transport equations. The so-called particle-in-cell (PIC) method is the Lagrangian algorithm originally proposed by Harlow, 1955 for solving compressible fluid flow. Its applicability was then extended to different kinds of flow (Grigoryev et al., 2002). One of the biggest advantages of this method is negligible numerical diffusion (Andrews and O'Rourke, 1996). The PIC method greatly helps to overcome difficulties related to modeling multiple phases in the flow. In particular, it allows tracking different materials, which is very important when modeling hydraulic fracturing to predict placement of different proppants and degradation of fluids and additives which, in turn, affects proppant placement.