1 Introduction
The relative permeability of the different fluids in a two-phase flow has been a lively area of study in scientific and engineering fields concerned with two-phase flows in geological reservoirs, such as in carbon capture and storage (CCS) fields, enhanced oil recovery (EOR) operations, geothermal power systems, and geological radioactive waste disposal repositories (Benson et al., 2015; C. Chen & Zhang, 2010; Gudjonsdottir et al., 2015; Niibori et al., 2011; Shad et al., 2008; Wu & Wang, 2020). Relative permeability is a crucial hydraulic property for modeling the flow of both fluids and assessing the mechanisms of fluid displacement in the reservoir. For example, when CO2 is injected into a CCS reservoir, the CO2 (nonwetting phase) displaces the existing fluid in the reservoir, such as oil or brine (wetting phase). Relative permeability values can be used to estimate the reduction in CO2 fluid flow due to surface-tension effects between CO2 and the brine, thus the parameter is useful to assess the injectivity of the CO2 (Benson et al., 2015; Burnside & Naylor, 2014). The relative permeability can also aid estimation of how much fluid can be displaced by CO2before the system reaches the wetting fluid irreducible saturation condition, limiting the CO2 volume that can be stored in the reservoir (Burnside & Naylor, 2014). Conversely, in EOR systems the best results are obtained when the relative permeability is high for oil and low for the injected fluid (Heins et al., 2014).
In a two-phase flow system, the interaction between the two fluids is vital in determining the relative permeability of each fluid. Therefore, the relative permeability value of each fluid is not only a function of saturation, but is also affected by other parameters related to its interaction with the other fluid component. Lenormand et al. (1988) reported that two parameters, viscosity ratio and capillary number, can explain the interaction between two immiscible fluids. The viscosity ratio (M ) is a dimensionless parameter describing the ratio between the viscosity of the injected nonwetting fluid and the viscosity of the ambient wetting fluid:
\begin{equation} M=\frac{\mu_{\text{nw}}\ }{\mu_{w}}\ \ \ \ \ \ \ (1)\nonumber \\ \end{equation}
where μ nw is the dynamic viscosity of the nonwetting fluid and μ w is the dynamic viscosity of the wetting fluid.
The capillary number (Ca ) is a dimensionless parameter describing the ratio between the viscous drag forces and the interfacial tension forces between two immiscible fluids:
\begin{equation} Ca=\frac{\mu_{\text{nw}}V_{\text{nw}}}{\sigma}\text{\ \ \ \ \ \ \ \ \ }(2)\nonumber \\ \end{equation}
where V nw is the average fluid velocity of the nonwetting fluid and σ is the interfacial tension (IFT) between the two fluids.
Despite the recognition that the relative permeability k is a function of M , several studies of this relationship have reported divergent results. An experimental study (Odeh, 1959) found that the relative permeability of the nonwetting fluid (k nw) increases and the relative permeability of the wetting fluid (k w) stays relatively constant as M increases, as have several other experimental and numerical studies (Dou & Zhou, 2013; Goldsmith & Mason, 1963; Huang & Lu, 2009; Jeong et al., 2017; Mahmoudi et al., 2017; Yiotis et al., 2007; Zhao et al., 2017). An analytical study of co-current annular flow in which the wetting fluid is distributed on the pore surface and the nonwetting fluid is in the middle of the pore produced empirical equations for the nonwetting and wetting fluids as a function of M and saturation:
\begin{equation} k_{\text{nw}}=S_{\text{nw}}\ [\frac{3}{2}M+{S_{\text{nw}}}^{2}(1-\frac{3}{2}M)]\ \ \ \ \ (3)\nonumber \\ \end{equation}\begin{equation} k_{w}=\frac{1}{2}{{(1-S}_{w})}^{2}(3-S_{w})\ \ \ \ \ \ (4)\nonumber \\ \end{equation}
where S nw is the saturation of the nonwetting fluid, and S w = 1 − S nw is the saturation of the wetting fluid. These equations suggest thatk w is not affected by increasing M and is a function of saturation alone. However, other studies have reported that k nw increases and k wdecreases as M increases (Ahmadlouydarab et al., 2012; Fan et al., 2019; Goel et al., 2016; Ramstad et al., 2010); thus, there is as yet no general agreement on the variation of k wwith increasing M . One of the challenges of previous studies was the difficulty of removing the effects of capillary forces and wettability factors when evaluating this relationship.
Similarly, studies of the influence of Ca on relative permeability is still imperfectly known. An experimental study (Fulcher et al., 1985) concluded that ks nw is a function of IFT rather than Ca , such that k nwincreases as IFT decreases (Ca increases), and thatk w consistently increases as Ca increases, but not as much as k nw increases, as IFT decreases. Several studies (Asar & Handy, 1989; Fan et al., 2019; Harbert, 1983) also found that k nw andk w increase as IFT decreases because the two fluids interfere less with each other and thus tend to form more well-connected flow pathways. One of the studies (Asar & Handy, 1989) showed that both fluids relative permeability curves tend to straighten and approach the 45° tangent line as IFT approaches zero. Other studies (Amaefule & Handy, 1982; Jiang et al., 2014; Shen et al., 2010) also concluded that both k nw andk w decrease as Ca decreases. On the other hand, a numerical study (Zhao et al., 2017) concluded thatk w increases with increasing Ca under neutral wetting conditions (θ = 90°) but stays relatively constant with increasing Ca under strong wetting conditions (θ = 135°). These divergent results warrant further investigations of how relative permeability changes with changing IFT and Ca . One of the challenges in this evaluation is the difficulty of isolating the effect of IFT while keeping other parameters, such as viscosity, constant. Thus, it is difficult to hold Ca constant in all simulation conditions.
Previous studies have usually evaluated the separate effects of Mand Ca on relative permeability. However, in two-phase flows, the effects of viscosity gradient and interfacial tension must be evaluated simultaneously to accurately predict the relative permeability in the system. Because few studies have evaluated the condition when both parameters influence relative permeability using a steady-state simulation, our aim in this work was to fill that knowledge gap.
In this paper, we clarify the effects of viscosity ratio and capillary number on the relative permeability of nonwetting and wetting fluids from the results of a color gradient lattice Boltzmann method (LBM) simulation using a three-dimensional (3D) digital rock. We begin by evaluating the effects of the two parameters individually by holding other parameters constant. We use the Euler–Poincaré characteristic or Euler number to describe the fluid connectedness, a factor that is directly related to relative permeability. We then mapk nw in various MCa conditions in search of general trends. By understanding the variations of relative permeability in the MCa parameter space, we can conduct accurate large-scale reservoir simulations by considering the reservoir conditions M and Ca and thereby contribute to the wide range of research regarding applications that employ two-phase fluid mixtures.