Numerical simulation of proppant transport in hydraulic fractures

A central issue in hydraulic fracturing treatment in petroleum wells is the transport of proppant particles by the injection fluid. In this paper, we present an innovative proppant transport model in a fixed rectangular- and elliptic-shaped slots. The proposed model is an improvement to the current modeling of proppant transport by applying a non-oscillatory numerical scheme which has high accuracy everywhere in solution domain, even close to the steep gradients. In addition, inertia, fracture wall, and concentration effects on proppant settling along with slurry evolution as a function of proppant concentration has been considered. This paper introduces the mathematical equations that govern the proppant transport phenomenon and discusses special front capturing numerical techniques, boundary conditions, coupling between proppant and slurry mass conservation equations and time stepping restrictions required for the solution stability. We incorporated published correlations obtained from proppant transport laboratory experiments in our numerical model to better capture the physics of the problem. 5th- order WENO scheme was used to avoid oscillation and diffusion at the proppant front since traditional finite difference discretization was found to be insufficient in solving the hyperbolic transport partial differential equations. Results show that the technique used in this study can capture the proppant distribution with minimum oscillation and diffusion. A series of sensitivity analysis was conducted to explore the legitimacy of these assumptions and to provide guidelines that allow more accurate predictions of the proppant and fluid transfer. Numerical results are presented to show how proppant distribution is impacted by the injection fluid viscosity, density difference between proppant particles and injection fluid, proppant size, and fluid flow injection rate. Results of the sensitivity analysis illustrate the significance of choosing appropriate viscosity of the injection fluid as small changes in the viscosity may cause noticeable effects on the concentration distribution. In addition, we found that variation of proppant size and density within a reasonable range have a modest effect on proppant concentration distribution. Furthermore, we also investigated the amount of gravity driven vertical motion of proppant which is driven by density differences (convection) and compare it to a second gravity driven motion which is proppant settlement. Both of these two well recognized mechanisms can occur inside a fracture during proppant placement, however, the importance of each mechanism as a function of proppant injection design parameters is not fully understood.