A New Model of Trapping and Relative Permeability Hysteresis for All Wettability Characteristics
- Elizabeth J. Spiteri (Chevron Corp.) | Ruben Juanes (Massachusetts Inst. of Tech.) | Martin J. Blunt (Imperial College) | Franklin M. Orr (Stanford University)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- September 2008
- Document Type
- Journal Paper
- 277 - 288
- 2008. Society of Petroleum Engineers
- 5.1.8 Seismic Modelling, 4.3.1 Hydrates, 5.2.1 Phase Behavior and PVT Measurements, 4.3.4 Scale, 5.4.2 Gas Injection Methods, 5.4 Enhanced Recovery, 5.5 Reservoir Simulation, 5.3.4 Reduction of Residual Oil Saturation, 5.4.1 Waterflooding, 5.1.1 Exploration, Development, Structural Geology, 5.3.1 Flow in Porous Media, 1.2.3 Rock properties, 5.5.1 Simulator Development, 5.3.2 Multiphase Flow
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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. 2000).
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 probed experimentally.
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.
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