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Abstract
Understanding and predicting the production behavior of naturally fractured reservoirs requires understanding flow in a single fracture. Individual fractures can be represented as two-dimensional networks of locations of wide and narrow aperture. Percolation theory prohibits two-phase flow in such a network under restricted conditions: that is, the relative-permeability curves cross at a relative permeability of zero. Using the Effective Medium Approximation (EMA), we illustrate the effect of aperture distribution and of gravity on these kri curves. In the absence of gravity effects, as the aperture distribution becomes broader, the cross-over point in the relative permeability curves approaches zero wetting-phase saturation. On the other hand, straight-line relative-permeability curves obtain if gravity segregation dominates flow. This can occur, however, only if both phases are free to attain gravity equilibrium, which may not be possible given the restriction on local two-phase flow. In fractures, viscous forces become significant at pressure gradients orders of magnitude lower than in rock matrix. Viscous effects would tend to lead to straight-line relative permeability curves, as well. We present also several simple models for non-Newtonian single-phase flow through fractures. Shear-thinning non-Newtonian fluids make the fracture appear to be wider than in Newtonian flow. In addition, plugging a fracture requires a higher fluid yield stress than expected from characterization of fracture aperture from Newtonian flow. Both non-Newtonian effects are greatest for fractures with broad aperture distributions.
Introduction
Naturally fractured oil and gas reservoirs occur worldwide (Reiss, 1980; Van Golf-Racht, 1982; Nelson, 1985). These reservoirs possess, in addition to intergranular porosity, a network of fractures that increases effective reservoir permeability by orders of magnitude. These fractures are not uniform slits with parallel walls, but are themselves two-dimensional networks of variable aperture (gap width) (Tsang, 1984; Brown and Scholz, 1985; Wang and Narasimhan, 1985; Brown and Kranz, 1986; Schrauf and Evans, 1986; Pyrak-Nolte et al., 1988; Morrow et al., 1989). Thus, in effect, fractures are 2D analogs of the 3D networks of throats and bodies that compose the pore network of rock matrix. This report examines the implications of this fact for laminar two-phase Newtonian flow (relative permeabilities) and single-phase non-Newtonian flow in fractures. The extension of this analysis to two-phase non-Newtonian flow would be straight-forward. The pore space of a fracture differs from the pore network of rock matrix in two important ways. First, a fracture is a two-dimensional network of apertures and constrictions. Percolation theory indicates that simultaneous multiphase flow is impossible in a spatially isotropic and uncorrelated 2D network if flow is dominated by capillary forces. (Or, more precisely (Fisher, 1961), in site percolation on a 2D network, one cannot have two or more distinct, overlapping, infinite clusters of connected pores as required for simultaneous multiphase flow under capillary-dominated conditions.) This theoretical result suggests that while conductivity of a fracture to single-phase flow may be large, conductivity to multiple phases may be much, much lower. The implications for overall reservoir productivity could be significant. Pruess and Tsang (1990) discuss the limited, and contradictory, available data on fracture relative permeabilities (see also Persoff et al., 1991). Most fractured-reservoir simulators assume straight-line relative permeabilities (krw = Sw, kmw = Snw), in sharp contrast to the prediction of percolation theory. One can identify four cases in which the percolation result would not apply:A film of wetting fluid could flow along fracture walls while non-wetting fluid occupies and flows through the fracture interior.Two phases could flow alternately, with the saturation fluctuating on either side of the percolation threshold (Persoff et al., 1991). Unless the fluctuations were large, however, the time-average relative permeability of both phases would still be near zero.If systematic spatial segregation occurs, for instance under gravity, or due to long-range, anisotropic auto correlation of the aperture distribution (Pruess and Tsang, 1990), then each phase could flow through the region where it predominates.If viscous or pressure forces are great enough to mobilize finite ganglia or "blobs" of a given phase, then two-phase flow could occur. In rock matrix, one can mobilize residual phases either by applying massive pressure gradients or by reducing capillary forces using surfactants (Larson et al., 1981; Chatzis and Morrow, 1981).
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