Gas and water relative permeability can be effectively modeled in many porous media using the modified Corey (1954) equations:

k_rg = (1 – (S_w-S_wc,g)/(1-S_gc-S_wc,g))^p (1-((S_w-S_wc,g)/(1-S_wc,g))^q)                                                                            (1)
k_rw = ((S_w-S_wc)/(1-S_wc))^r                                                                                                                                         (2)

        where S_w= water saturation, S_gc = critical gas saturation (expressed as fraction gas saturation), S_wc,g = critical water saturation for gas equation (expressed as fraction water saturation), S_wc = critical water saturation, and p, q, and r are exponents reflecting pore size distribution and architecture. These variables change with such lithologic variables as grain size, sorting, and volume, type, and distribution, of diagenetic clay and secondary porosity. Beyond influences of lithologic variables within a homogeneous rock, lithologic anisotropy such as bedding planes parallel or perpendicular to flow will affect relative permeability. Corey and Rathjens (1956) showed that bedding planes perpendicular to flow tend to increase S_gc and that parallel bedding tends to decrease S_gc and cause inflections in both the krg and krw curves. In higher permeability rocks (i.e., k > 10 md) the measurement and operational definitions of S_gc and S_wc can be clearly defined within a few saturation percent. However, as permeability decreases and rocks move more into the transition zone interval of the capillary pressure curve, the nature of both becomes more complex with implications for modeling using relative permeability.

        Based on comparison of wireline log-measured water saturations and the lack of significant water production for low-permeability reservoirs, it is generally thought that many low-permeability sandstones are at or near “irreducible” water saturation (S_wi) or critical water saturation (S_wc). “Irreducible” saturation can be operationally defined as the saturation at which further increase in capillary pressure does not result in a “significant” decrease in water saturation. Critical water saturation can be operationally defined as the saturation at which water is immobile or water flow is negligible on the time scale of importance for the evaluation of flow properties. Critical water saturation is also often experimentally defined as the saturation at which the ratio of the nonwetting phase flow to water (wetting phase) flow is greater than 1000 (i.e. water flow represents less than 0.001 of total flow). Many western low-permeability reservoirs produce gas water-free or at very low water production rates indicating that water saturations are at or near S_wc or that the water relative permeability (k_rw) is low. Critical gas saturation (S_gc) represents the saturation below which the gas phase is discontinuous and therefore does not flow. Experimentally this is defined as the saturation at which a threshold pressure achieves first detectable gas flow.

        For higher permeability rocks S_wi and S_wc can be similar and for clean sandstones are often between 10-20%. Because of this low water saturation the effective gas permeability at or near “irreducible” water saturation (k_eg,Swi) is within 75-100% of the absolute single-phase permeability (k_i). However, in low-permeability rocks, k_eg,Swi can be significantly less than k_i because water occupies critical pore space even at or near S_wc. Byrnes (2003) compiled low-permeability sandstone relative permeability data and showed that with relative permeability referenced to the in situ Klinkenberg gas permeability (k_ik) the gas relative permeability curve could be modeled using the following parameters:

S_wc,g = 0.16 + 0.053*logk_ik (where if k_ik < 0.001 md then S_wc,g = 0)                                                                        (3)
S_gc = 0.15 - 0.05*logk_ik                                                                                                                                             (4)
p = 1.7                                                                                                                                                                        (5)
q = 2                                                                                                                                                                           (6)

        where S_wc,g and S_gc are expressed in fractions and k_ik is expressed in md. In Figure 1 the bounding Corey-equation curves represent curves with S_wc,g and S_gc values for rocks with permeability of 0.001 md and 1 md, which approximate the range of permeabilities of the samples for which relative permeability curves are shown. Equation 3 models S_wc,g = 0.16 for k_ik = 1 md and approaches S_wc,g = 0 at k_ik = 0.001 md. Conversely, S_gc = 0.15 for k_ik = 1 md and approaches S_gc = 0.30 at k_ik = 0.001 md. It is important to note that S_wc,g calculated using Equation 3 is relevant only to the influence of water saturation on krg and does not correspond to critical water saturation relative to water flow (i.e., S_wc). S_wc,g defines the water saturation below which krg remains ~ 1. In contrast, S_wc defines the water saturation below which krw is approximately zero. The S_gc and S_wc terms are defined separately for each phase to allow calculation of relative permeability curves at saturations below those for which flow is unmeasurable or negligible.

        Assuming the Corey equations presented here are approximately correct, Equation 4 can be interpreted to indicate that S_gc increases with decreasing absolute permeability. Conventionally, for higher permeability rocks, S_gc ranges from ~5-15% and is not highly dependent on permeability. All the data presented here can be interpreted to indicate that S_gc is greater than this in low-permeability sandstones, unless the samples have fractures or high-permeability laminae, which would provide flow paths with negligible saturation change. It is possible to postulate two aspects of pore architecture that explain the relationship of increasing S_gc with decreasing ki. In higher permeability rocks S_gc might be lower because the fraction of pores needed to be occupied by gas to achieve a percolation threshold (i.e. a single connective path through the pore network) is low since there are many pores with a significant fraction of the total pore volume that do not need to be gas saturated to achieve the percolation threshold. Conversely, in low-permeability rocks with thin sheet-like pores interconnecting a limited number of large pore bodies, that represent the majority of the pore space, if a connective path encountered just a few pore bodies, the filling of these would represent a greater fraction of the pore volume and consequently a higher S_gc. In addition, lower permeability rocks tend to have greater bedding complexity. As Corey and Rathjens (1956) reported, this increased heterogeneity could also be responsible for increasing S_gc with decreasing k_ik. Alternately, it is possible to model the observed krg curves with a similar constant low value of S_gc for all k_ik but with the exponent, p, changing with k_ik. This would imply that pore architecture and size distribution change with decreasing permeability.

        To test the above models, threshold mercury injection capillary pressure measurements were performed coupled with electrical resistivity measurements to determine S_gc. This experimental methodology allowed the observation of the threshold capillary pressure and saturation at which a connective path was established as observed by an abrupt increase (several orders of magnitude) in electrical conductivity across the sample as mercury reached the end of the core providing a conductive path from one end of the core to the other. This electrical conductivity technique is highly sensitive to connectivity. These measurements confirm that Snwc (S_gc) increases with decreasing k_ik but Snwc also varied as a function of rock lithology, which was not handled in equation 4. This implies that both S_gc and p change with decreasing permeability and in response to lithology.

        Equation 3 implies that krg at any given S_w increases with increasing absolute permeability, that is, the krg curves shift up to higher values of krg as k_ik increases or, alternately, krg curves shift to higher values of S_w as k_ik increases. The first explanation can be considered to represent the idea that in higher permeability rock at any given saturation gas occupies larger pores in a higher permeability sample compared to a lower-permeability sample and therefore the gas relative permeability is greater. The second explanation can be considered to represent the condition that in a high permeability sample higher water saturation is possible for the same krg because the water occupies smaller pores that are less influential or are inconsequential to flow. Both mechanisms are valid. Higher permeability samples can exhibit krg values at low-moderate S_w (~0-0.2) that are 75-100% of the dry gas permeability k_ik at S_w=0. This can result from water occupying pore space at S_w < S_wc that is inconsequential to flow and therefore when gas occupies these same pores its relative additional contribution to gas flow is also negligible. Equation 3 predicts S_wc>0.20 for higher permeability rocks (k_ik>10 md) with S_wc approaching zero with decreasing k_ik down to 0.001 md and below. Equation 4 predicts S_gc<0.1 for higher permeability rocks and approaches S_gc = 0.30 for rocks of k_ik=0.001 md. Taken together these trends imply that as absolute permeability decreases, and pores become progressively more sheet-like, any water in the pore space interferes with gas flow and that greater gas saturation is needed to establish a connective path. These two conditions are consistent. Perhaps most important is that values for S_w at S_gc in lower-permeability rocks begin to approach saturations present in the reservoir. This would indicate that very low-permeability rocks have gas saturation that it is nearly immobile on the time frame relevant to commercial production. Commercially viable gas production from intervals with these properties requires the presence of a high permeability “carrier” channel (ie., fracture or high permeability bed) within close proximity to decrease flow distance and therefore time.

        Jones and Owens (1981) first reported that water permeability is progressively less than Klinkenberg gas permeability with decreasing permeability for k_ik < 1 md. Ward and Morrow (1987) proposed the modification of the Corey water relative permeability equation 2, shown above, to calculate water relative permeability with values presented relative to k_ik by using the ratio of the k_w/k_ik:

k_rw = ((S_w-S_wc)/(1-S_wc))4 (k_w/k_ik)                                                                                                                     (7)

        Because of the experimental difficulty of measuring krw in very low permeability rocks little work is published. In addition, because of the significant difference in the mobility (k/m; permeability/viscosity) of gas and water, gas/water flow ratios of 1000/1 are achieved at high water saturations (i.e., high S_wc). At a limit S_wc~ Swi, but definition of S_wi is problematic in low-permeability rocks because rocks with k_ik< 0.1 md are still in the transition zone for gas column heights of hundreds of feet. Curves generated using higher, flow-measured, S_wc values are useful for short-term flow prediction but can deviate from observed flow over longer time periods. As with krg, krw curves can be modeled either with a constant exponent, r, and variable S_wc that increases with decreases k_ik, or with a constant S_wc and changing value of r with decreasing k_ik. High pressure laboratory gas-water drainage experiments indicate that water is mobile to low saturations at high displacement pressures. This indicates that although increasing S_wc with decreasing k_ik effectively models short-term flow, models involving r as a function of k_ik may more accurately represent water relative permeability behavior.

References:

Byrnes, A.P., 2003, “Aspects of permeability, capillary pressure, and relative permeability properties and distribution in low-permeability rocks important to evaluation, damage, and stimulation”, Proceedings Rocky Mountain Association of Geologists – Petroleum Systems and Reservoirs of Southwest Wyoming Symposium, Denver, Colorado, September 19, 2003, 12 p.

Corey, A.T., 1954, “The interrelation between gas and oil relative permeabilities”, Producers Monthly, vol. 19, no. 1, November, p. 38-41.

Corey, A.T., and Rathjens, C.H., 1956, “effect of stratification on relative permeability”, J. of Petroleum Technology, Transactions, Am. Inst. Of Mining Engrs., Technical Note 393, December, p. 69-71.

Jones, F.O., and Owens, W. W., 1980, “A laboratory study of low-permeability gas sands”, Journal of Petroleum Technology, vol. 32, no. 9, p. 1631-1644.

Ward, J.S., and Morrow, N.R., 1987, “Capillary pressure and gas relative permeabilities of low permeability sandstone”, Soc. of Petroleum Engineers Formation Evaluation, Sept., p. 345-356.

Figure 1. Gas relative permeability curves for various western sandstones from Byrnes (2003). The bounding black curves represent curves constructed using equations 3 through 6 for rocks of 0.001 md and 1 md, representing the approximate upper and low limits of the samples for which curves are shown. S_gc = 0.3 and 0.15 for k_ik = 0.001 md and 1 md, respectively. Bounding red curves represent Eq. 3 with S_gc = 0.1 and p = 2.8 for k_ik=0.001md and p=2 for k_ik =1 md, respectively.

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