Modeling Immiscible Two‐Phase Flow in Rough Fractures From Capillary to Viscous Fingering

Abstract

We develop an efficient computational model for simulating fluid invasion patterns emerging in variable aperture fractures. This two‐dimensional model takes into account the effect of capillary force on the fluid‐fluid interfaces and viscous pressure drop in both fluid phases. The pressure distribution is solved at each time step based on mass balance and local cubic law, considering an imposed pressure jump condition at the fluid‐fluid interface. This pressure jump corresponds to the Laplace pressure which includes both terms related to the out‐of‐plane (aperture‐spanning) curvature and to the in‐plane curvature. Simulating a configuration that emulates viscous fingering in two‐dimensional random porous media confirms that the model accounts properly for the role of viscous forces. Furthermore, direct comparison with previously obtained experimental results shows that the model reproduces the observed drainage patterns in a rough fracture reasonably well. The evolutions of tip location, the inlet pressures, and the invading phase fractal dimensions are analyzed to characterize the transition from capillary fingering to viscous fingering regimes. A radial injection scenario of immiscible invasion is also studied with varying modified capillary number and viscosity ratio, showing displacement patterns ranging from capillary fingering to viscous fingering to stable displacement. Such simulations using two contact angles show that the invading phase becomes more compact when the wetting condition changes from strong to weak drainage, as already observed in 2‐D porous media. The model can be used to bridge the gap in spatial scales of two‐phase flow between pore‐scale modeling approaches and the continuum Darcy‐scale models.

Gravitational and Finite‐Size Effects On Pressure Saturation Curves During Drainage

“…To overcome this shortcoming, microscopic approaches such as Stokes' solution for flow in single pores with a specified geometry or lattice Boltzmann methods (LBM) have been investigated to reproduce experiments (Aursjø et al, 2010;Fiorentino et al, 2017;Misztal et al, 2015). Similarly, pore-scale network models have been developed to study a wide range of displacement processes, including drainage and imbibition (Aker et al, 1998;Joekar Niasar et al, 2009;Kallel et al, 2017;Nordhaug et al, 2003;Nsir et al, 2012Nsir et al, , 2018Sheng & Thompson, 2013;Singh & Mohanty, 2003;Tørå et al, 2012;Yang et al, 2019). Ewing and Berkowitz (1998) developed a generalized growth model based on invasion percolation to simulate immiscible displacement in saturated porous media.…”

Mixing and finger morphologies in miscible non-Newtonian solution displacement

“…Immiscible fluid displacement, where surface tension between the two fluids results in so-called capillary forces acting at the fluid-fluid interface, has been studied for Newtonian fluids in both linear (Bonn et al, 1995;Chevalier et al, 2006;Miranda & Widom, 1998;Saffman & Taylor, 1958) and radial (J. Den Chen, 1989;Dias et al, 2012;Miranda & Widom, 1998;Thomé et al, 1989;White & Ward, 2014;Yang et al, 2019) Hele-Shaw cells. Miscible Newtonian fluids, where mixing between the injected and displaced fluids takes place during the displacement, gives rise to more irregular fingering patterns than what is observed for immiscible fluids (J.…”

Roughness Control on Multiphase Flow in Rock Fractures

“…The roughness induces the aperture variability between the two rock surfaces (AlQuaimi & Rossen, 2017;Brown, 1987), determining the invasion path (Lee et al, 2010;Neuweiler et al, 2004), and controls the displacement patterns and phase distribution (Babadagli et al, 2015;Karpyn et al, 2007). A number of experimental (Chen et al, 2017; and numerical studies (Chen, Guo, et al, 2018;Yang et al, 2019) confirm that the main mechanism controlling the displacement patterns in rough fractures is similar to that in porous media (Toussaint et al, 2012), and similar empirical phase diagrams of displacement patterns have been proposed as a function of capillary number Ca and viscosity ratio M. However, given the significant differences in flow geometry for rock fractures and porous materials, the phase diagram originally established for porous media may not be applicable to flow in rough fractures (Babadagli et al, 2015;. It has been discovered that the roughness of rock surfaces can induce localized flow channels even when a more viscous fluid displaces a less viscous one ), which has not been observed in experiments of Hele-Shaw type or in porous media.…”