Emulsion Flow through a Packed Granular Material with Multiple Drop Breakup

We study the flow of an emulsion of immiscible deformable drops through a randomly packed granular material. This problem is relevant to many applications, including oil filtration through underground reservoirs. Of particular interest is to obtain the pressure gradient - flow rate relationships and delineate the conditions when the emulsion can no longer squeeze through the material due to drop blockage in the pores by capillary forces; it is also intriguing to study the effects of drop breakup in this process. If the emulsion drops were much smaller than the

pores, some effective-medium model could be assumed for the emulsion. However, in most cases, the drops are comparable in size with the particles and pores, and the system has to be handled as a three-phase one, by direct numerical

simulation, with distinct velocities for the drop- and continuous phases. A recent multipole-accelerated boundary-integral algorithm [1] has been extended and systematically used in this work to study emulsion flow

through granular materials with different microstructures. The material skeleton is modeled as a random arrangement of many monodisperse solid spheres rigidly held in a periodic cell in mechanical equilibrium under the action of contact forces. Two extreme cases are of interest (i) frictionless spheres forming a "random close packing" (RCP), with the average density of 0.637 and the coordination number of six, and (ii) highly-frictional particles in a 'random loose packing' (RLP), with the density of packing 0.55 and coordination number of four (within the percolating cluster transmitting the stress). An emulsion of many non-wetting drops flows under a specified volume-averaged pressure gradient. Even with only 50-100 particles and drops in a periodic cell, the problem presents very severe computational difficulties, because of necessary superhigh resolution (~ 10 000 boundary elements per surface) and a large number ( ~10 000 -50 000) of time steps to evaluate long-time averages for the phase velocities. Multipole acceleration, built in our algorithm, has a two-order-of-magnitude advantage over the standard boundary-integral coding at each time step, and appears to

be, at present, the only way to perform such simulations. Compared to the original algorithm [1], the present version incorporates topological mesh changes on drop surfaces

(with the help of techniques from [2]) to simulate a cascade of multiple drop breakups observed in this problem at sufficiently large capillary numbers. A novel defragmentation algorithm is used to continue simulations

after each pinch-off. Due to geometrical constraints imposed by solid particles, the drop shapes are typically rather compact at breakup, and the daughter drops are comparable in size; small satellite drops are almost never observed. Another significant extension, compared to [1], is ensemble

averaging of the results over many realizations of a granular material. We found that using the empirical Carman-Kozeny correlation for the pure-fluid velocity in the definition for the capillary number significantly reduces the dispersion of data to facilitate ensemble averaging. For drop-to-medium viscosity ratio of unity, drop volume fraction of 20% in the entire space, and nondeformed drop-to-particle size ratio of ~0.5, the average drop- and continuous phase permeabilities are studied as functions of the capillary number, both for RLP and RCP structures; the critical thresholds for squeezing (vs. trapping) to occur are evaluated. The effect of the size ratio is also explored. Generally, the kinematic drop-phase velocity is higher than that for the continuous phase, except for near-critical squeezing conditions. For general viscosity ratios, the boundary-integral iterations for the solid phase are no longer decoupled, which makes the ensemble averaging a formidable task. Nevertheless, in this case, we obtained similar results for individual material realizations.

[1] Zinchenko A.Z., Davis R.H. 2008 Algorithm for direct numerical simulation

of emulsion flow through a granular material. J. Comput. Phys., vol. 227,

pp.7841-7888.

[2] V. Cristini, J. Blawzdziewicz, M. Loewenberg 2001 An adaptive mesh

algorithm for evolving surfaces: simulations of drop breakup and coalescence.

J. Comput. Phys., vol. 168, pp. 445-463.