Imperial College Consortium on Pore - Scale Modelling Second Progress Report

We describe the early stages of a poreto coreto reservoir-scale investigation of wettability variation and its impact on flow during oil recovery. We use a threedimensional pore-scale network representation of a Berea sandstone to predict relative permeability and capillary pressure hysteresis. We successfully match experimental data for the water-wet case, and then focus upon the effect of variations in initial water saturation associated with capillary rise above the oil-water contact. This may lead to wettability variations with height, because the number of pore-walls which may be rendered oil-wet during primary drainage increases as the oil saturation increases. We investigate the validity of empirical hysteresis models in which ‘scanning curves’ are used to connect bounding imbibition and drainage curves for a given initial (minimum) water saturation. If the wettability varies with initial water saturation, we demonstrate that the bounding imbibition curve, which is measured from the lowest water saturation (and hence the most oil-wet conditions), does not yield the correct scanning curves at higher water saturations. We then use a conventional simulator in conjunction with the relative permeability curves obtained from the network model to investigate the largescale impact of wettability variations on waterflood efficiency. Our results demonstrate that the proper inclusion of hysteresis is important to correctly predict recovery if wettability varies with height above the oil-water contact. Assuming that the reservoir is uniformly water-wet or oil-wet, or using empirical hysteresis models, leads to an underestimate of recovery. Imperial College Consortium on Pore-Scale Modelling December 2001 49 Introduction The wettability of an oil/water/rock system can have a significant impact on flow during oil recovery, and upon the volume and distribution of the residual oil (e.g. Salathiel, 1973; Morrow et al., 1986; Morrow, 1990; Jadhundan and Morrow, 1995). Wettability depends upon factors such as the mineralogy of the rock, the composition of the oil and water, the initial water saturation, and the temperature (e.g. Dubey and Waxman, 1991; Dubey and Doe, 1993; Wolcott et al., 1993; Buckley, 1995). Several studies have successfully used network models to investigate the effect of wettability variations on flow at the pore-scale (e.g. Heiba et al., 1983; McDougall and Sorbie, 1995; Blunt, 1997a; 1997b; Dixit et al., 1997; 1999; Hui and Blunt, 2000); the aim of this study is to investigate and predict the effect of wettability variations on flow at the coreand reservoir-scales using network models in conjunction with conventional simulations. The study forms part of a larger project in which multiphase flow processes at the pore-, coreand reservoir-scales are upscaled dynamically using the appropriate simulation technique at each scale (Jackson and Blunt, 2000; in press). In this paper, we describe the early stages of the study. We use a three-dimensional (3D) network model of a Berea sandstone to investigate relative permeability and capillary pressure hysteresis. The model is reconstructed directly from a sample of the sandstone, so the pore-size distribution and co-ordination number are fixed and are not ‘tuned’ to match experimental data (Bakke and Øren, 1997; Øren et al., 1998). This makes the network model more likely to be truly predictive. Before primary drainage, the rock is assumed to be strongly water-wet. Following drainage, wettability variations are modelled by changing the advancing contact angle assigned to each oil-filled pore. Different pores may have different contact angles. We successfully match experimental data for the water-wet case (Oak, 1990), then focus upon the effect of variations in initial water saturation associated with capillary rise above the oil-water contact (OWC). This may lead to wettability variations with height, because the number of pore-walls which may be rendered oil-wet during primary drainage increases as the oil saturation increases (e.g. Jerauld, 1996a; 1996b). We investigate the validity of empirical hysteresis models in which ‘scanning curves’ are used to connect bounding imbibition and drainage curves for a given initial (minimum) water saturation (Killough, 1976; Carlson, 1981), and demonstrate that if the wettability varies with initial water saturation, the bounding imbibition curve, which is measured from the lowest water saturation (and hence most oil-wet conditions), does not yield the correct scanning curves at higher water saturations. We then use a conventional simulator in conjunction with the relative permeability curves obtained from the network model to investigate the large-scale impact of wettability variations on waterflood efficiency, and demonstrate that the proper inclusion of hysteresis is important to correctly predict recovery if wettability varies with height above the OWC. Network model The 3-D network model is a cube of volume 9mm, containing 12349 pores and 26146 throats, reconstructed directly from a sample of Berea sandstone (Bakke and Øren, 1997; Øren et al., 1998) (Fig. 1). Each pore and throat is represented as a duct with a triangular cross-section, characterized by an inscribed radius which controls the threshold capillary entry pressure, effective corner angles which control the amount of Imperial College Consortium on Pore-Scale Modelling December 2001 50 fluid held in wetting layers, and an effective volume which controls the mobile (nonclay bound) saturation (Blunt, 1997a; 1997b; Zhou et al., 1997; Firincioglu et al., 1999; Patzek, 2000). Empirical formulae are used to compute the hydraulic conductance of each pore and throat (Zhou et al., 1997). Two-phase flow is simulated for primary drainage and imbibition assuming that capillary forces dominate, so the pores and throats are filled in order of increasing capillary entry pressure. This is reasonable for low capillary number flow (Hilfer and Øren, 1996; Blunt, 1997a). The drainage cycle begins with the network fully saturated with water and strongly water wet, with the receding contact angle θr = 0o. Oil then enters the network, and as the capillary pressure is increased step by step, invades the pore or throat with the lowest capillary entry pressure in an invasion percolation process (Wilkinson and Williamson, 1983; Dias and Wilkinson, 1986). At each step, water and oil saturations, relative permeabilities, and the capillary pressure, are calculated subject to pressure boundary conditions at the inlet and outlet faces, and periodic boundaries on the other faces. To avoid end-effects, only a subset of the network model is included. Drainage ends when a target capillary pressure or saturation has been reached, or when all pores and throats have been invaded by oil. Wettability variations are modelled by changing the advancing contact angle θa assigned to each oil-filled pore. Different pores may have different contact angles. Depending upon the number of pores and throats filled with oil, and the range of advancing contact angles, this approach allows us to model a mixed wet system, in which only those pores invaded by oil become oil-wet, a fractionally wet system, in which a fraction of the pores and throats invaded by oil become oil-wet, or a system which is both mixed and fractionally wet. Spontaneous and/or forced imbibition is then simulated, including piston type displacements, cooperative pore body filling, layer flow and snap-off (e.g. Lenormand Figure 1. Reconstructed pore space, from Oren . (1998). et al Imperial College Consortium on Pore-Scale Modelling December 2001 51 et al., 1988; Blunt, 1997b). Saturations, relative permeabilities and capillary pressures are calculated in the same way as for drainage. Imbibition ends when a target capillary pressure or saturation has been reached, or when all available pores and throats have been invaded by water. Relative permeability and capillary pressure hysteresis Relative permeability and capillary pressure typically exhibit hysteresis during drainage and imbibition, and several workers have argued that this can have a significant impact on flow during recovery (e.g. Land, 1968; Killough, 1976; Carlson, 1981; Kossack, 2000). Killough (1976) and Carlson (1981) presented models for hysteresis in which relative permeabilities and capillary pressures can vary between drainage and imbibition via intermediate ‘scanning’ curves (Fig. 2). The drainage and imbibition curves which bound the scanning curves are determined experimentally, and the scanning curves are obtained by interpolating or re-mapping the bounding curves. Each scanning curve corresponds to a reversal in the direction of saturation change. The first set of scanning curves correspond to a reversal from drainage to imbibition, which occurs at the maximum wetting phase saturation obtained after drainage. In a water-wet reservoir, these scanning curves correspond to different initial water saturations. The methods for obtaining the scanning curves are described in Killough (1976) and Carlson (1981), and will not be reproduced here. Figure 3 shows schematically the regions in which the scanning curves are located for both models. Figure 2. Hysteresis in relative permeability. Solid lines denote experimentally determined bounding curves; dashed lines denote scanning curves. After Killough (1976). See Equations (1) and (2) for an explanation of the terms on the left-hand plot. R el at iv e pe rm ea bi lit y Wetting phase saturation Drainage