A volumetric-smoothed particle hydrodynamics based Eulerian-Lagrangian framework for simulating proppant transport

Different numerical methods have been applied to simulate the proppant transport in petroleum engineering, which can be roughly categorized as the Eulerian-Eulerian and Eulerian-Lagrangian models. Recently, a hybrid Eulerian-Lagrangian (E-L) approach, the multiphase particle-in-cell (MP-PIC) method, has been successfully applied to model large-scale proppant transport problems by introducing the concept of parcels (clusters of particle). In the MP-PIC method, particle-particle interaction force is expressed as the gradient of particle stress. The calculation of this gradient strongly depends on the interpolation between particle properties and Eulerian grids, which could lead to problems such as non-physical particle suspension, particle agglomeration and nonconserved interparticle interactions. In this study, a new method, the volumetric-smoothed particle hydrodynamics (V-SPH) method, is proposed to improve the calculation accuracy of the particle-particle interaction forces in the original MP-PIC method. In the V-SPH method, the calculation of the particle stress gradient no longer depends on the background Eulerian grids and the conservation of the interparticle stress is also guaranteed. In this paper, detailed introduction of the V-SPH based Eulerian-Lagrangian framework is provided. The reliability of the proposed V-SPH method is validated against both the numerical and experimental results in literature. By comparing with the original MP-PIC method, we observe that the non-physical particle agglomeration, as well as non-physical particle suspension problems can be well solved with the proposed new model. In addition, the impact of some key parameters in the V-SPH method on simulation results are also investigated. The choice of the PPP (number of particles per parcel) and the treatment of boundary particle deficiency are found to play important roles in model accuracy and efficiency. The V-SPH method proposed in this work can provide more accurate results than the original MP-PIC method, especially in the regions of dense proppant concentration, with comparable computing efficiency. With proper treatment of boundary deficiencies, it is promising to be used in more complex field-scale proppant transport problems.