Finite volume simulations of particle-laden viscoelastic fluid flows: application to hydraulic fracture processes

Accurately resolving the coupled momentum transfer between the liquid and solid phases of complex fluids is a fundamental problem in multiphase transport processes, such as hydraulic fracture operations. Specifically we need to characterize the dependence of the normalized average fluid-particle force $\langle F \rangle$ on the volume fraction of the dispersed solid phase and on the rheology of the complex fluid matrix. Here we use direct numerical simulations (DNS) to study the creeping flow ($Re\ll 1$) of viscoelastic fluids through static random arrays of monodisperse spherical particles using a finite volume Navier-Stokes/Cauchy momentum solver. The numerical study consists of $N=150$ different systems, in which the normalized average fluid-particle force $\langle F \rangle$ is obtained as a function of the volume fraction $\phi$ $(0 < \phi \leq 0.2)$ of the dispersed solid phase and the Weissenberg number $Wi$ $(0 \leq Wi \leq 4)$. From these predictions a closure law $\langle F \rangle(Wi,\phi)$ for the drag force is derived for the quasi-linear Oldroyd-B viscoelastic fluid model which is, on average, within $5.7\%$ of the DNS results. Additionally, a flow solver able to couple Eulerian and Lagrangian phases is developed, which incorporates the viscoelastic nature of the continuum phase and the closed-form drag law. Two case studies were simulated using this solver, in order to assess the accuracy and robustness of the newly-developed approach for handling particle-laden viscoelastic flow configurations with $O(10^5-10^6)$ rigid spheres that are representative of hydraulic fracture operations.