This paper presents a new and comprehensive model for proppant transport that is applicable inside both the wellbore and the fracture,along with an appropriate fast computational algorithm.This model can be used efficiently in both real-time and design modes of hydraulic fracturing processes. The spatio-temporal proppant evolution is crucial in hydraulic fracturing processes as it directly affects production during shut-in and flowback.Yet,the computational cost is quite high because of multidi- mensional multiphase flow models,while most of the numerical approximations,such as the standard Galerkin finite element,are limited in their validity,prohibiting the numerical models from properly modeling the proppant dynamics correctly and practically. This paper discusses a new one-dimensional(1D)two-phase flow model that accounts for the leading two-dimensional(2D)effects,such as proppant settling,resuspension,and bed transport,among others. Accounting for these effects helps ensure the conservation of momentum and mass conservation for the fluid flow for the proppant as well as for the proppant bed transport.The momentum conservation equations embed the interphase momentum transfer between the fluid and the solid phases,the momentum changes caused by proppant settling and resuspension,and the momentum exchange associated with the shear on the dynamic built-up proppant bed.This is accompanied by the mass conservations equations to include the fluid and the proppant mass flux induced by the settling and resuspension. This computational model simulates proppant transport by using a first-order implicit scheme in time and first-order upwind scheme in space for the convection over a collocated grid for flow variables.The simulations demonstrate that the proppant bed is gradually formed and expanded downstream.These simulations also show that the formation of the proppant bed is caused mainly by proppant settling,and height change is determined by interactions among proppant settling and its resuspension during transport. Also,it is shown that the proppant properties strongly affect slurry transport and proppant bed dynamics. The general results provide insight into the different mechanisms of the transport process,including gravitational settling and resuspension of proppant and proppant bed buildup and transport inside both the wellbore and fractures.This model provides a more pragmatic approach that enables field engineers to gain better insight into the proppant transport phenomena during a fracturing treatment.
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