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

Abstract

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, parameterized through the Weissenberg number Wi measuring the relative magnitude of elastic to viscous stresses in the fluid. 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 $\phi$ $(0 < \phi \le 0.2)$ of the dispersed solid phase and the Weissenberg number Wi $(0 \le Wi \le 4)$. From these predictions a closure law $\langle F(\phi,Wi) \rangle$ for the drag force is derived for the quasi-linear Oldroyd-B viscoelastic fluid model (with fixed retardation ratio $\beta = 0.5$) which is, on average, within $5.7\%$ of the DNS results. In addition, a flow solver able to couple Eulerian and Lagrangian phases (in which the particulate phase is modeled by the discrete particle method (DPM)) 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, 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. Three-dimensional settling processes in a Newtonian fluid and in a quasi-linear Oldroyd-B viscoelastic fluid are both investigated using a rectangular channel and an annular pipe domain. Good agreement is obtained for the particle distribution measured in a Newtonian fluid, when comparing numerical results with experimental data. For the cases in which the continuous fluid phase is viscoelastic we compute the evolution in the velocity fields and predicted particle distributions are presented at different elasticity numbers $0\le El \le 30$ (where $El=Wi/Re$) and for different suspension particle volume fractions.

Appendix A

In the DNS study presented in Sect. 4, we considered two different domain configurations, one with spheres having centroids in a wall region of thickness a around all four lateral edges of the flow domain and another in which the sphere centroids are excluded from this wall region. We refer to these cases as the no-excluded volume and excluded volume configurations, respectively. As shown in Fig. 18a when spheres are allowed to be located in the wall region (blue color), i.e., when their centroid is located less than one radius from the wall, then the boundary acts as a perfectly periodic wall. In the opposite case the boundary walls exclude the spheres and act like rigid stress-free periodic walls. In fact, based on Fig. 18b, we can calculate the probability of a single sphere being located in the wall region. Assuming a square cross section with a width of 8a, the total cross-sectional area is $64a^2$. Regarding the blue annular area, i.e., the region which excludes the spheres near the wall, the area is $(64a^2-36a^2)=28a^2$. Hence, the probability of a randomly placed sphere being located in the excluded region is equal to $28a^2/64a^2=0.4375$ and the overall/fraction area of spheres (of volume fraction $\phi$) in this region can be as large as $0.4375\phi$. Therefore, as $\phi$ increases it is important that when particles are randomly distributed in the domain they should be allowed to be placed with centroids near the walls.

Fig. 18

Schematic representation of (a) excluded and no-excluded volume regions and (b) area fraction in the excluded volume region

In Fig. 19, we show contours of the velocity magnitude in the transverse $y-z$ plane for configurations with an excluded volume and no-excluded volume region. From the distribution of velocity magnitude contours, it can be seen that in the excluded volume configuration (i.e., Fig. 19a), the larger local concentration of rigid impenetrable spheres in the middle of the square channel pushes the strongest fluid flow outwards towards the walls causing a stagnant region near the channel center. On the other hand, in the no-excluded volume configuration (i.e., Fig. 19b), the fluid flow is more evenly distributed across the entire channel. This weakly affects the average drag force exerted on the spheres as shown in Table 1 and Fig. 3.

Fig. 19

Steady flow field around one representative random array of particles in a channel filled with Newtonian fluid. Contours of the velocity magnitude field $\Vert \mathbf{u} \Vert$ (with the axial direction pointed out of the plane of the page) are represented for particle volume fraction $\phi =0.2$, in the $y-z$ plane with (a) excluded volume near the walls and (b) a no-excluded volume configuration. In the latter case the velocity field is more evenly distributed across the entire cross section of the domain