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Accurately resolving the coupled momentum transfer between the liquid andsolid phases of complex fluids is a fundamental problem in multiphase transportprocesses, such as hydraulic fracture operations. Specifically we need tocharacterize the dependence of the normalized average fluid-particle force$\langle F \rangle$ on the volume fraction of the dispersed solid phase and onthe rheology of the complex fluid matrix. Here we use direct numericalsimulations (DNS) to study the creeping flow ($Re\ll 1$) of viscoelastic fluidsthrough static random arrays of monodisperse spherical particles using a finitevolume 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-Bviscoelastic fluid model which is, on average, within $5.7\%$ of the DNSresults. Additionally, a flow solver able to couple Eulerian and Lagrangianphases is developed, which incorporates the viscoelastic nature of thecontinuum phase and the closed-form drag law. Two case studies were simulatedusing this solver, in order to assess the accuracy and robustness of thenewly-developed approach for handling particle-laden viscoelastic flowconfigurations with $O(10^5-10^6)$ rigid spheres that are representative ofhydraulic fracture operations.

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