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 M –Ca conditions in
search of general trends. By understanding the variations of relative
permeability in the M –Ca 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.