Relative Permeability From Capillary Pressure - OnePetro
However, by providing a capillary pressure correlation, we are able to extract is restricted to a single-component system, practical examples may be found in. examples of capillary pressure measurements proposed a correlation between permeability. Capillary pressure relationships are dimensional (Capillary pressure is often defined as the.
For this example, water is the wetting phase, and gas is the nonwetting phase. As shown in Figs. Wettability of a solid with respect to two phases is characterized by the contact angle. Popular terminology for saturation changes in porous media reflects wettability: Water is the wetting phase. Gas does not penetrate the medium in Fig. As capillary pressure increases beyond this value, the saturation of the water continues to decrease. Morrow and Melrose  argue that capillary pressure measurements have not reached equilibrium if the capillary pressure trend asymptotically approaches an irreducible water saturation.
As the water saturation decreases during a measurement, the capacity for flow of water rapidly diminishes, so the time needed for equilibration often increases beyond practical limitations. Hence, a difference develops between the measured relationship and the hypothetical equilibrium relationship, as shown in Fig. After completing measurements of capillary pressure for primary drainage, the direction of saturation change can be reversed, and another capillary pressure relationship can be measured—it is usually called an imbibition relationship.
Imbibition is often analogous to the waterflooding process. The primary drainage and imbibition relationships generally differ significantly, as shown in Fig. This difference is called capillary pressure hysteresis—the magnitude of capillary pressure depends on the saturation and the direction of saturation change.
For imbibition of a strongly wetting phase, the capillary pressure generally does not reach zero until the wetting-phase saturation is large, as shown in Fig. For a less strongly wetting phase, the capillary pressure reaches zero at a lower saturation, as shown in Fig. Capillary pressure behavior for secondary drainage is also shown in Figs. The endpoint saturation of a phase for a specific displacement process depends on the structure of the porous material, the wettabilities with respect to the various phases, the previous saturation history of the phases, and the extent of the displacement process the number of pore volumes injected.
The endpoint saturation also can depend on IFTs when they are very low, and on the rate of displacement when it is very high. Results reported by Chatzis et al. These results suggest two general conclusions. First, the residual saturation of a wetting phase is less than the residual saturation of a nonwetting phase.
Capillary pressure -
Second, the residual saturation of a nonwetting phase is much more sensitive to heterogeneities in the porous structure. This wide range of wetting possibilities is an obstacle to interpreting or predicting the effect of wettability on endpoint saturations.
Indeed, conflicting results for different porous media are likely. For example, Jadhunandan and Morrow  report that residual oil saturation displays a minimum value for mixed-wet media as wettability shifts from water-wet to oil-wet—counter to the results of Bethel and Calhoun,  who reported a maximum for media of uniform wettability. In the subsections below, specific relationships for endpoints of the oil, gas, and water phases are discussed. Interest in the mobility of condensates in retrograde gas reservoirs developed in the s, as it was observed that condensates could hamper gas production severely in some reservoirs, particularly those with low permeability.
The trend of increasing critical condensate saturations with decreasing permeability, as summarized by Barnum et al. Trapped, or Residual, Gas Saturation. As shown in Fig. The relationship of Fig. For example, in a gas reservoir, encroachment of the aquifer will lead to trapping of some portion of the gas. Several correlations and summaries for residual gas saturation are found in the literature. Katz and Lee  provide a summary of residual gas saturations in a graphical form that is useful for estimates.
According to the model presented by Naar and Henderson  for multiphase flow through rock, the trapped or residual gas saturation is one-half of its initial saturation; this Naar-Henderson rule is the simplest correlation for residual gas. Agarwal  correlated a large collection of residual gas saturations for consolidated and unconsolidated sandstones, for unconsolidated sands, and for limestones.
The ranges of parameters in the correlations are summarized in Table The correlations may be erroneous outside of these ranges.
Relative permeability and capillary pressure
Three of the Agarwal correlations are listed below: Permeability k is in millidarcies. Residual oil saturations after waterflooding or gasflooding are clearly significant for oil recovery. Here, the dependence of residual oil saturation on initial oil saturation and capillary number for a waterflood will be considered.
The relationship between initial and residual oil saturation is termed the oil-trapping relationship. For strongly water-wet rocks, the oil-trapping relationship should be identical to the gas-trapping relationship.
Indeed, because of this analogy and because it is easier to measure gas-trapping relationships, few oil-trapping relationships have been measured. A set of oil-trapping relationships reported by Pickell et al.
Oil-trapping relationships are important for estimating reserves in transition zones.
In conventional reservoir engineering, residual oil saturation refers to the remaining oil saturation after a displacement that starts near the maximum initial oil saturation, which generally equals one minus the initial water saturation. This topic has received much more attention in the literature than oil-trapping functions. The capillary number is the ratio of viscous forces to capillary forces.
It is represented quantitatively with various expressions, as summarized by Lake. A popular definition of the capillary number is as follows: The capillary number is small less than 0. The following example shows just how small capillary numbers can be. Capillary forces do indeed dominate flow processes for waterfloods. Even in high-velocity regions, such as the vicinity of a well that is producing oil and water, the capillary number will remain very small.
Having defined the capillary number, the relationship between residual oil saturation and capillary number will be discussed next. As the capillary number for an oil-displacing process increases, residual oil saturation decreases in the manner sketched in Fig. Above the "critical capillary number," the rate of decrease of Sor is particularly rapid. The critical capillary number is 10—5 to 10—4 for porous media with fairly uniform pore sizes.
With increasing distribution of pore sizes, the critical capillary number decreases, the Sor at low Nc increases, and the domain for decreasing S or becomes broader.Porosity determination using mercury intrusion
Extensive discussion of these relationships is available elsewhere. Residual Irreducible Water Saturation. Residual, or irreducible, water saturation Swi is the lowest water saturation that can be achieved by a displacement process, and it varies with the nature of the process—gas displacement or oil displacement.
Also, Swi varies with the extent of the displacement, as measured by pore volumes of oil or gas injected or by time allowed for drainage. To be more specific, the results of Chatzis et al. Furthermore, Swi should increase slightly with increasing breadth of grain-size distribution. Significant variations in Swi should occur when small clusters of consolidated media of one grain size are surrounded by media of another grain size. If the grains of the clusters are smaller than those of the surrounding media, Swi increases; if the grains of the clusters are larger than those of the surrounding media, Swi decreases.
The saturation of water in an oil or gas reservoir at discovery is called the connate water saturation, or Swc. The connate water saturation and the irreducible water saturation can differ.
If the reservoir processes that produced the connate water saturation can be replicated, then the Swi for the replicated processes should be the same as Swc. Swc is significant for its connection to initial oil or gas saturation in a reservoir. The connate water saturation will also affect initial oil or gas relative permeability and, hence, the economic viability of a reservoir. Bulnes and Fitting  concluded that low-permeability limestone reservoirs are more viable than sandstone reservoirs of the same permeability because the connate water saturation is lower in the limestones than in the sandstones; as a result, the relative permeabilities to oil are higher in the limestones than in the sandstones.
Salathiel  observed that the connate water saturations in carefully retrieved rock samples from some oil reservoirs are substantially lower than can be achieved when the rock is waterflooded and then oilflooded. He attributed this effect to the mixed-wettability condition.
When the reservoir was first invaded by oil, the rock was water-wet, and low water saturations were obtained. However, the wettability of the rock surfaces that were now in contact with oil changed from water-wet to oil-wet as portions of the hydrocarbons adsorbed onto the solid surfaces.
The need for upscaling should diminish as increases in computer power permit higher-resolution models. To obtain accurate measurements of capillary pressure and relative permeabilities, tests with representative samples at representative conditions are critical.
Much of the available data in our industry do not pass this standard. Capillary end effects and viscous fingering have corrupted a significant portion of relative permeability data.
If capillary pressure and relative permeabilities are available, the extent of this corruption for a sample can be assessed and sometimes corrected. Such interpretations are particularly susceptible to error caused by the heterogeneity of the sample used for measurements.
This susceptibility was conceded in the original literature on wettability interpretation, but it is not widely acknowledged. The quality of estimates of capillary pressure and relative permeability with network models is increasing.