The complex physics of multiphase flow in porous media are usually modeled
at the field scale using Darcy-type formulations. The key descriptors of such
models are the relative permeabilities to each of the flowing phases. It is
well known that, whenever the fluid saturations undergo a cyclic process,
relative permeabilities display hysteresis effects.
In this paper, we investigate hysteresis in the relative permeability of the
hydrocarbon phase in a two-phase system. We propose a new model of trapping and
waterflood relative permeability, which is applicable for the entire range of
rock wettability conditions. The proposed formulation overcomes some of the
limitations of existing trapping and relative permeability models. The new
model is validated by means of pore-network simulation of primary drainage and
waterflooding. We study the dependence of trapped (residual) hydrocarbon
saturation and waterflood relative permeability on several fluid/rock
properties, most notably the wettability and the initial water saturation. The
new model is able to capture two key features of the observed behavior: (1)
non-monotonicity of the initial-residual curves, which implies that waterflood
relative permeabilities cross; and (2) convexity of the waterflood relative
permeability curves for oil-wet media caused by layer flow of oil.
Hysteresis refers to irreversibility or path dependence. In multiphase flow,
it manifests itself through the dependence of relative permeabilities and
capillary pressures on the saturation path and saturation history. From the
point of view of pore-scale processes, hysteresis has at least two sources:
contact angle hysteresis, and trapping of the nonwetting phase.
The first step in characterizing relative permeability hysteresis is the
ability to capture the amount of oil that is trapped during any displacement
sequence. Indeed, a trapping model is the crux of any hysteresis model: it
determines the endpoint saturation of the hydrocarbon relative permeability
curve during waterflooding.
Extensive experimental and theoretical work has focused on the mechanisms
that control trapping during multiphase flow in porous media (Geffen et al.
1951; Lenormand et al. 1983; Chatzis et al. 1983). Of particular interest to us
is the influence of wettability on the residual hydrocarbon saturation. Early
experiments in uniformly wetted systems suggested that waterflood efficiency
decreases with increasing oil-wet characteristics (Donaldson et al. 1969; Owens
and Archer 1971). These experiments were performed on cores whose wettability
was altered artificially, and the results need to be interpreted carefully for
two reasons: (1) reservoirs do not have uniform wettability, and the fraction
of oil-wet pores is a function of the topology of the porous medium and initial
water saturation (Kovscek et al. 1993); and (2) the coreflood experiments were
not performed for a long enough time, and not enough pore volumes were injected
to drain the remaining oil layers to achieve ultimate residual oil saturation.
In other coreflood experiments, in which many pore volumes were injected, the
observed trapped/residual saturation did not follow a monotonic trend as a
function of wettability, and was actually lowest for intermediate-wet to
oil-wet rocks (Kennedy et al. 1955; Moore and Slobod 1956; Amott 1959).
Jadhunandan and Morrow (1995) performed a comprehensive experimental study of
the effects of wettability on waterflood recovery, showing that maximum oil
recovery was achieved at intermediate-wet conditions.
An empirical trapping model typically relates the trapped (residual)
hydrocarbon saturation to the maximum hydrocarbon saturation; that is, the
hydrocarbon saturation at flow reversal. In the context of waterflooding, a
trapping model defines the ultimate residual oil saturation as a function of
the initial water saturation. The most widely used trapping model is that of
Land (1968). It is a single-parameter model, and constitutes the basis for a
number of relative permeability hysteresis models. Other trapping models are
those of Jerauld (1997a) and Carlson (1981). These models are suitable for
their specific applications but, as we show in this paper, they have limited
applicability to intermediate-wet and oil-wet media.
Land (1968) pioneered the definition of a "flowing saturation," and
proposed to estimate the imbibition relative permeability at a given
actual saturation as the drainage relative permeability evaluated at a
modeled flowing saturation. Land’s imbibition model (1968) gives
accurate predictions for water-wet media (Land 1971), but fails to capture
essential trends when the porous medium is weakly or strongly wetting to oil.
The two-phase hysteresis models that are typically used in reservoir simulators
are those by Carlson (1981) and Killough (1976). A three-phase hysteresis model
that accounts for essential physics during cyclic flooding was proposed by
Larsen and Skauge (1998). These models have been evaluated in terms of their
ability to reproduce experimental data (Element et al. 2003; Spiteri and Juanes
2006), and their impact in reservoir simulation of water-alternating-gas
injection (Spiteri and Juanes 2006; Kossack 2000). Other models are those by
Lenhard and Parker (1987), Jerauld (1997a), and Blunt (2000). More recently,
hysteresis models have been proposed specifically for porous media of mixed
wettability (Lenhard and Oostrom 1998; Moulu et al. 1999; Egermann et al.
All of the hysteresis models described require a bounding drainage curve and
either a waterflood curve as input, or a calculated waterflood curve using
Land’s model. The task of experimentally determining the bounding waterflood
curves from core samples is arduous, and the development of an empirical model
that is applicable to non-water-wet media is desirable. In this paper, we
introduce a relative permeability hysteresis model that does not require a
bounding waterflood curve, and whose parameters may be correlated to rock
properties such as wettability and pore structure.
Because it is difficult to probe the full range of relative permeability
hysteresis for different wettabilities experimentally, we use a numerical
tool--pore-scale modeling--to predict the trends in residual saturation and
relative permeability. As we discuss later, pore-scale modeling is currently
able to predict recoveries and relative permeabilities for media of different
wettability reliably (Dixit et al. 1999; Øren and Bakke 2003; Jackson et al.
2003; Valvatne and Blunt 2004; Al-Futaisi and Patzek 2003, 2004). We will use
these predictions as a starting point to explore the behavior beyond the range
In summary, this paper presents a new model of trapping and waterflood
relative permeability, which is able to capture the behavior predicted by
pore-network simulations for the entire range of wettability conditions.
© 2008. Society of Petroleum Engineers
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- Original manuscript received:
21 July 2005
- Meeting paper published:
9 October 2004
- Revised manuscript received:
11 April 2007
- Manuscript approved:
23 January 2008
- Version of record:
20 September 2008