Effective Rheology of Two-Phase Flow in a Capillary Fiber Bundle Model

Abstract

We investigate the effective rheology of two-phase flow in a bundle of parallel capillary tubes carrying two immiscible fluids under an external pressure drop. The diameter of the tubes vary along the length which introduce capillary threshold pressures. We demonstrate through analytical calculations that a transition from a linear Darcy to a non-linear behavior occurs while decreasing the pressure drop ∆P , where the total flow rate Q varies with ∆P with an exponent 2 as Q ∼ ∆P 2 for uniform threshold distribution. The exponent changes when a lower cut-off Pm is introduced in the threshold distribution and in the limit where ∆P approaches Pm, the flow rate scales as Q ∼ (|∆P | − Pm) 3/2 . While considering threshold distribution with a power α, we find that the exponent γ for the non-linear regime vary as γ = α + 1 for Pm = 0 and γ = α + 1/2 for Pm > 0. We provide numerical results in support of our analytical findings. PACS numbers:Understanding the hydrodynamic properties of simultaneous flow of two or more immiscible fluids is essential due its relevance to a wide variety of different systems in industrial, geophysical and medical sectors [1,2]. Different applications, such as bubble generation in microfluidics, blood flow in capillary vessels, catalyst supports used in the automotive industry, transport in fuel cells, oil recovery, ground water management and CO 2 sequestration, deal with the flow of bubble trains in different types of systems, ranging from single capillaries to more complex porous media. The underlying physical mechanisms in multiphase flow are controlled by a number of factors, such as the capillary forces at the interfaces, viscosity contrast between the fluids, wettability and geometry of the system, which make the flow properties different compared to single phase flow. When one immiscible fluid invades a porous medium filled with another fluid, different types of transient flow patterns, namely viscous fingering [3,4], stable displacement [5] and capillary fingering [6] are observed while tuning the physical parameters [7]. These transient flow patterns were modeled by invasion percolation [8] and diffusion limited aggregation (DLA) models [9]. When steady state sets in after the initial instabilities, the flow properties in are characterized by relations between the global quantities such as flow rate, pressure drop and fluid saturation [10,11]. It has been observed theoretically and experimentally that, in the regime where capillary forces compete with the viscous forces, the two-phase flow rate of Newtonian fluids in the steady state no longer obeys the linear Darcy law [12,13] but varies as a power law with the applied pressure drop [14][15][16][17]. Tallakstad et al. [14,15] experimentally measured the exponent of the power law to be close to two (= 1/0.54) in a two-dimensional system and followed this observation up with arguments why the exponent should be two. Rassi et al. [16] found a value for the exponent varying between 2.2 (= 1/0.45) and 3.0 (= 1/0.33) in a three-dimensio...

Rheology of Immiscible Two-phase Flow in Mixed Wet Porous Media: Dynamic Pore Network Model and Capillary Fiber Bundle Model Results

“…Stated more in detail, the investigations in this work have been carried out by, firstly, calculating the total volumetric flow rate in a model consisting of a bundle of capillary tubes with mixed wet properties (Roy et al 2019;Sinha et al 2013). Thereafter, case studies with various specific wetting angle distribution have been performed through numerical calculations which confirmed the analytical results in addition to providing a holistic picture.…”

“…In this case, the volumetric flow rate scales nonlinearly with the pressure drop due to the fact that increasing the pressure drop by a small amount creates new connecting paths in addition to increase the flow in the previously connected paths. Earlier works (Roy et al 2019;Sinha et al 2021;Tallakstad et al 2009a;Rassi et al 2011;Tallakstad et al 2009b;Aursjø et al 2014;Gao et al 2020a;Zhang et al 2021) have provided experimental, theoretical and numerical evidences for this phenomena in porous media under uniform wetting conditions. Instead of assuming uniform wetting conditions, we here investigate the same phenomena using non-uniform wetting conditions, theoretically and numerically.…”

Role of Pore-Size Distribution on Effective Rheology of Two-Phase Flow in Porous Media

“…In this section, we will study the analytically solvable capillary fiber bundle model (CFBM) (Scheidegger, 1953(Scheidegger, , 1974, which can be considered as a prototype for a one-dimensional porous medium. The model was recently studied by the present authors to explore the non-linearity in the effective rheology in two-phase flow in a bundle of capillary tubes (Roy et al, 2019). Here we will analytically derive the relation between average flow rate and pressure drop for this model for different distributions of poreradii.…”

“…(3) Note that the above theoretical approaches find the exponent in the non-linear regime β to be equal to 2, thus hinting at universality. Recently, Roy et al (2019) have studied the effect of the threshold distribution on the effective rheology of two Newtonian fluids in a capillary bundle model. The model consists of a bundle of parallel capillary tubes with variable diameters along their lengths which introduce thresholds for each tube (Scheidegger, 1953(Scheidegger, , 1974.…”

Effective Rheology of Bi-viscous Non-newtonian Fluids in Porous Media

“…We now consider an analytically solvable model for the flow. Let us assume that the network consists of a set of parallel links placed between two fluid reservoirs kept at pressure p = 0 and p = ∇p < 0, i.e., we are describing the capillary fiber bundle model [14][15][16]. The constitutive equation for the fiber bundle is then given by…”