A Lagrangian Approach to Modelling Proppant Transport with Tip Screen-Out in KGD Hydraulic Fractures

This study introduces a continuum approach to model proppant transport in hydraulic fractures in a Lagrangian frame of reference. The model for the proppant transport is based on the recently obtained slurry flow solution inside a channel, where the latter utilizes a phenomenological constitutive relationship for a slurry. This approach allows us to describe the transition from Poiseuille flow with an effective viscosity to Darcy flow as the particle concentration increases towards the maximum value. The algorithm is presented for the one-dimensional case, for which propagation of a symmetric Kristinovich–Zheltov–Geertsma–De Klerk fracture is considered. To examine the effectiveness of the Lagrangian approach for proppant transport modelling, a set of parameters, for which proppant particles reach the fracture tip and cause the development of a proppant plug is selected. In this situation, the coupling between the hydraulic fracture propagation and proppant transport is the most significant. To estimate the accuracy of the Lagrangian proppant transport model, the results are compared to the predictions of an Eulerian proppant transport model, which utilizes the same algorithm for hydraulic fracture propagation. It is shown that, although both approaches have the same convergence rate, the error of the Lagrangian approach is three to five times smaller, which depends on the number of proppant elements used in the Lagrangian approach. This permits us to use a coarser mesh for hydraulic fracture propagation, and to obtain results with similar accuracy up to a hundred times faster.