Numerical simulation of proppant transport in propagating fractures with the multi-phase particle-in-cell method

In this work, the proppant transport process in large-scale propagating fractures is simulated using an Eulerian–Lagrangian method. Fracture propagation is solved using the Perkins–Kern–Nordgren(PKN) model, while the fluid-particle system is solved with the multi-phase particle-in-cell (MP-PIC) method. The fluid motion is governed by volume-averaged Navier–Stokes equations, and solved using the finite volume method, and the particle motion is solved by applying Newton’s second law in a Lagrangian manner. Based on the original MP-PIC method, an extended 2D system of governing equations for fluid-particle flow is derived to solve the moving boundary problems associated with fracture propagation. By means of this method, the fluid-particle interaction is fully coupled, and the propagating fracture is considered as a prior-known boundary for the fluid and particle phases. Several numerical experiments are performed to validate the method for simulating fluid motion and proppant settling behaviors in a fracture through comparison with results in the literature. The simulation results of the 2D framework are also compared with those of 3D framework and show a good agreement. Large-scale problems of proppant transport in propagating fractures for different proppant and fracturing fluid properties, including the leak-off effect, are then simulated using this method. The Lagrangian feature of the MP-PIC method allows for flexible design of proppant injection, such as injection of proppant with multi-densities and/or multi-sizes.