Modeling of Interfacial Instability, Conductivity and Particle Migration in Confined Flows

Daihui Lu, Purdue University

Abstract

This thesis analyzed three fundamental fluid dynamics problems arising from multiphase flows that may be encountered in hydraulically fractured flow passages. During hydraulic fracturing (“fracking”), complex fluids laden with proppants are pumped into tight rock formations. Flow passages in these formation are naturally heterogeneous with geometric variations, which become even more pronounced due to fracking. Upon increasing the flow area (and, thus, the conductivity of the rock), crude oil, shale gas or other hydrocarbons can then flow out of the formation more easily. In this context, we encounter the following three fluid mechanical phenomena: fluid–fluid interfacial instabilities, flow-wise variation of the hydraulic conductivity, and particle migration in the pumped fluids. First, we studied the (in)stability of the interface between two immiscible liquids in angled (tapered) Hele-Shaw cells, as model of a non-uniform flow passage. We derived an expression for the growth rate of perturbations to the flat interface and for the critical capillary number, as functions of the small gap gradient (taper). On this basis, we formulated a three-regime theory to describe the interface’s stability. Specifically, we found a new regime in which the growth rate changes from negative to positive (converging cells), or from positive to negative (diverging cells), thus the interface’s stability can change type at some location in the cell. We conducted three-dimensional OpenFOAM® simulations of the Navier–Stokes equations, using the continuous surface force method, to validate the theory. Next, we investigated the flow-wise variation of the hydraulic conductivity inside a nonuniformly shaped fracture with permeable walls. Using lubrication theory for viscous flow, in conjunction with the Beavers–Joseph–Saffman boundary condition at the permeable walls, we obtained an analytical expression for the velocity profile, conductivity, and wall permeation velocity. The new expression highlights the effects of geometric variation, the permeability of the walls, and the effect of flow inertia. The theory was validated against OpenFOAM® simulations of the Navier–Stokes equations subject to a tensorial slip boundary condition. Finally, we extended the utility of phenomenological models for particle migration in shear flow using the physics-informed neural networks (PINNs) approach. We first verified the approach for solving the inverse problem of radial particle migration in a non-Brownian suspension in an annular Couette flow. Then, we applied this approach to both non-Brownian and Brownian suspensions in Poiseuille slot flow, for which a definitive calibration of the phenomenological migration model has been lacking. Using PINNs, we identified the unknown/empirical parameters in the physical model, showing that (unlike assumptions made in the literature) they depend on the bulk volume fraction and shear Péclet number.

Degree

Ph.D.

Advisors

Christov, Purdue University.

Subject Area

Fluid mechanics|Physics|Artificial intelligence|Hydraulic engineering|Mathematics|Mechanics|Petroleum engineering

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