Journal of Petroleum Science and Engineering

Transcription

1 Journal of Petroleum Science and Engineering 66 (2009) Contents lists available at ScienceDirect Journal of Petroleum Science and Engineering journal homepage: A generalized correlation for predicting gas condensate relative permeability at near wellbore conditions M. Jamiolahmady, M. Sohrabi, S. Ireland, P. Ghahri Institute of Petroleum Engineering, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, UK article info abstract Article history: Received 20 December 2006 Accepted 2 February 2009 Keywords: relative-permeability gas condensate coupling effect inertial effect fluid flow low interfacial fluid tension systems It is a well established finding, both experimentally and theoretically that relative permeability (k r )ofgas condensate systems at low interfacial tension (IFT) depends on velocity and IFT. There are a number of correlations available both in the literature and commercial reservoir simulators that account for the coupling (i.e., the increase of k r as velocity increases and/or IFT decreases) and inertial (i.e., the reduction of k r as velocity increases) effects at near wellbore conditions. These functional forms, which with the exception of Whitson et al. (Whitson, C.H., Fevang, O., Saevareid, A., Oct Gas condensate relative permeability for well calculations, SPE Proc. of SPE Annual Technical Conference and Exhibition Texas) are based on saturation, estimate the two effects separately and include a number of parameters, which should be determined by cumbersome and expensive k r measurements at low IFT and high velocity conditions. We have previously reported the development of a fractional flow based correlation (Jamiolahmady, M., Danesh, A., Henderson, G.D. and Tehrani, D.H., Dec Variations of gas condensate relative permeability with production rate at near wellbore conditions: a general correlation, SPE Reserv. Eng. Evalu. J., 9 (6), pp ), which expressed the combined effect of coupling and inertia simultaneously, but its dependency on fluid viscosity limited its use to the range of IFT values studied. In this paper we present a generalized correlation based on relative-permeability-ratio, which is closely related to a fractional flow of gas and condensate. The parameters of the new correlation are either universal, applicable to all types of rocks, or can be determined from commonly measured petrophysical data. The performance of the correlation has been evaluated over a relatively wide range of IFT and velocity variations. This has been conducted by comparing its prediction with the gas condensate relative permeability values measured on porous media with very different characteristics. These data had not been used in developing the correlation. The results are quite satisfactory confirming the generality of the correlation in providing reliable information on gas condensate relative permeability values for near wellbore conditions Elsevier B.V. All rights reserved. 1. Introduction The characteristics of gas and condensate flow, when pressure falls below dew point, are significantly different from those of conventional gas oil systems and accurate determination of gas condensate relative permeability (k r ) values, is a major challenge (Jamiolahmady et al., 2000). The effect of interfacial tension (IFT) on relative permeability at low IFT values has been known for a considerable time (e.g., Bardon and Longeron, 1980; Asar and Handy, 1988). The improvement to the relative permeability of condensing systems due to an increase in velocity is also a relatively well established experimental finding (e.g., Danesh et. al., 1994; Henderson et al., 1997, 2001; Ali et al., 1997; Blom et al., 1997). This improvement in relative permeability of low IFT systems as velocity increases and/or IFT decreases, known as the coupling effect, has been attributed to the simultaneous coupled flow Corresponding author. Tel.: ; fax: addresses: jami.ahmady@pet.hw.ac.uk (M. Jamiolahmady), mehran.sorabi@pet.hw.ac.uk (M. Sohrabi), mehran.sorabi@pet.hw.ac.uk (P. Ghahri). of the gas and condensate phases with the intermittent opening and closure of the gas passage by the condensate at the pore level (Jamiolahmady et al., 2000, 200). This cyclic two-phase flow pattern, unique to gas condensate systems, was observed in the micromodel experiments conducted by authors ((Jamiolahmady et al., 2000). The developed single pore mechanistic model capturing the competition between viscous and capillary forces, consistent with the experiments, showed that for these low IFT systems, there is a highly conductive film of condensate flowing with the gas. The condensate evolves at the pore throat and blocks the gas passage, after which the flow of gas continues till it overcomes the capillary barrier. As IFT increases the number of pores promoting this flow pattern gradually diminishes switching to a conventional Darcy type of flow mechanism used in the channel flow concept. Hence, when the effect of multiple pore interaction was included in a network of pores (Jamiolahmady et al., 200) k r values comparable with experimentally measured values were reported. There are now several correlations in the literature (e.g., Henderson et al.,1996; Blom and Hagoort,1998; Pope et al.,1998; Whitson et al.,1998) and in commercial reservoir simulators (e.g., ECLIPSE, VIP) to express the /$ see front matter 2009 Elsevier B.V. All rights reserved. doi: /j.petrol

2 M. Jamiolahmady et al. / Journal of Petroleum Science and Engineering 66 (2009) Nomenclature A j Parameter, j=1 to 7 C j Constant, j=1 to 15 k Absolute permeability, m 2 or md k(s wi ) Permeability at S wi,m 2 or md k e Effective permeability, m 2 or md k r Relative permeability, dimensionless k rgtr Relative permeability ratio, i.e., k rg /(k rg + k rc ), dimensionless (k rgtr ) Micpoeff k rgtr value at which k rg affected by the presence of micro-pores, dimensionless L Length, m P Pressure, Pa or psia Q Flow rate, m s 1 S wi Immobile water saturation, dimensionless S w1 μm Wetting phase (air) saturation corresponding to one micron radius from mercury porosimetry P c curve, dimensionless u Darcy velocity, m s 1 Y g Gas weight function for interpolation of k r, dimensionless (Y g ) Main Gas weight function for majority of data points following main trend, dimensionless (Y g ) Micpoeff Gas weight function for data points not following main trend due to effect of micro-pores, dimensionless Greek letters β Single-phase inertial factor, m 1 β(s wi ) Single-phase inertial factor at S wi,m 1 β g Two-phase inertial factor for gas phase, m 1 ϕ Porosity of porous medium, % μ Viscosity, kg (ms) 1 or cp ρ Density, kg m σ Interfacial tension, IFT, between gas and liquid, N m 1 or mn m 1 σ r Ratio of the base IFT of.0 mn m 1 to current IFT, dimensionless Subscript ave Average value of the quantity b Value of the quantity for the base case at highest measured interfacial tension, IFT= mn m 1, and lowest velocity g Gas phase iner Value of the quantity affected by inertia j An index c Condensate m Miscible case T Total Abbreviations f g Gas to total flow rate ratio IFT Interfacial tension Nc Capillary number, ratio of viscous to capillary forces Ncr Ratio of current Nc to base Nc Operators Absolute value Δ Difference operator Divergence operator Gradient operator capillary number (ratio of capillary to viscous forces) dependence of the two-phase flow of gas and condensate at these low IFT systems. Henderson et al. (2001), through some steady-state k r measurements, reported that at low condensate saturations the relative permeability values decreased when the flow rate was increased, due to the dominance of the inertial effect. However, the positive coupling effect surpassed the inertial effect at higher condensate saturation resulting in a net increase of relative permeability with velocity. There are some correlations to express the effect of inertia using a two-phase inertial factor β g (e.g., Henderson et al., 2001; Whitson et al., 1999; Mott et al., 2000; App and Mohanty, 2002). The main difficulty in accounting for the inertial effect in all the reported correlations is the requirement for estimating β g. Correlations have been reported in the literature for calculating β g (e.g., Narayanaswamy et al., 1999; Dacun, and Thomas, 2000), which relate β g mainly to S g and/or k rg. However, the application of these correlations to gas condensate systems is open to question as they have been developed mainly for gas water systems, with the immobile water phase (Al-Kharusi, 2000). We recently proposed a fractional flow based correlation (Jamiolahmady et al., 2006) accounting for the combined effects of coupling and inertia as a function of fractional flow. The choice of fractional flow as the main independent variable rather than saturation was based on our observation that it could express relative permeability of core samples with different characteristics more reliably. That is, different core samples have a different pore size distribution, which could lead to different distributions of the phases, i.e., different phase saturation, at similar flow conditions. Furthermore, fractional flow is directly related to fluid composition and pressure at steady-state conditions, which is generally prevailing near the wellbore, hence, it makes it practically much more attractive compared to that of saturation which depends on core characteristics. The condensate relative permeability is also linked to that of gas by the fractional flow, thereby eliminating the need for a separate correlation for its calculation. However, the dependency of relative permeability to fluid viscosity and the lack of a proper definition of the lower limit of the correlation limited its use to the range of IFT values studied. In other words, as will be discussed in Section 2., the reported base IFT of 0.85 mn m 1,abovewhichk r is not a function of IFT, is not a universal value and the threshold capillary number fixing the lower limit of velocity (below which k r is not a function of velocity) has also not been identified. It will be demonstrated that although fractional flow and relative permeability ratio are closely related, the inclusion of viscosity in the definition of independent variable makes it difficult to fix the base conditions. In other words, the base k r curve reported as a function of fractional flow rate makes it fluid dependent whilst its effect can easily be accounted for separately when calculating the relative permeability ratio from the known fractional flow and viscosity values facilitating its generality. This together with the lack of proper definition for base capillary number, as will be demonstrated in Section, could result in a highly erroneous estimation of relative permeability if fractional flow based correlation is used. In this manuscript we have removed both these limitations and sought simpler mathematical expressions. This enables us to have a correlation with either universal parameters, applicable to all types of rocks, or those that can be determined from commonly measured petrophysical data. 2. Structure of correlation In a two-phase flow system Darcy's law, which was developed for a single-phase flow, is used by replacing the absolute permeability (k) with the effective permeability (k e ), which is reported as relative permeability (k r =k e /k). In gas condensate systems the relative permeability could be affected by both the coupling (i.e., the increase of k r as velocity increases and/or IFT decreases) and inertial (i.e., the reduction of k r as velocity increases) effects. In the new proposed formulation, k rg is interpolated between the base (k rgb ) and the miscible-fluids curves (k rgm ) using an interpolation function

3 100 M. Jamiolahmady et al. / Journal of Petroleum Science and Engineering 66 (2009) Y g. This simple interpolation approach has been favoured by many investigators to express the coupling effect on k r of gas condensate systems at different values of capillary number (e.g., Coats, 1980; Henderson et. al., 1996; Whitson et al., 1999). However, here, similarly to the fractional flow based correlation (Jamiolahmady et al., 2006) we modify both k rgm and k rb for the effect of inertia, which enables us to express the combined effect of coupling and inertia simultaneously, using Eq. (1). k rg = Y g k 1 Y rgb g k ð1þ rgm iner + The complete list of variables can be found in the nomenclature section. The main independent variable is the relative-permeability-ratio defined as, k rgtr = k rg k rg + k rc = iner : 1 : ð2þ 1+ k rc k rg Solving the above equation for condensate relative permeability, k rc, gives: k rg 1 k rgtr k rc = : ðþ k rgtr Therefore, when k rg is determined as a function of k rgtr this equation automatically gives the corresponding value of k rc for the same value of k rgtr. k rgtr ratio could be calculated in terms of gas fractional flow (f g ), and fluid viscosities. That is, f g is defined as, f g = Q g Q g + Q c = u g u g + u c = h i k r μ h i k r μ + c h i k r μ g ; ð4þ where Q is the volumetric flow rate, u is the Darcy velocity, μ is the viscosity and subscripts (g) and (c) refers to the gas and condensate phases, respectively. The extended form of Darcy's law for two-phase flow, Eq. (5), has been used in Eq. (4) to relate velocity to relative permeability as, u= kk rjp j = g; c; ð5þ μ j where P is the pressure, k is the absolute permeability and is the gradient operator. Solving Eq. (4) for relative-permeability-ratio gives:!! k 1 f rc g μ = c : ð6þ k rg f g μ g Substituting Eq. (6) into Eq. (4), after some mathematical manipulation, k rgtr can be related to f g by, k rgtr = k rg μ g f = g : ð7þ k rc + k rg μ g f g + μ c 1 f g The use of relative-permeability-ratio based on the summation of relative permeability values of gas and condensate in the denominator of Eq. (7), rather than the relative permeability of one of the two phases (e.g., k rgcr, which is the gas to condensate relative permeability ratio) avoids the mathematical difficulty associated with the latter (i.e., k rgcr when k rc 0 which is equivalent to condensate flow rate 0) Miscible-fluids relative permeability for gas (k rgm ) In the present k rgtr based correlation the gas miscible curve is modified to include the inertial effect with an argument similar to that presented for the fractional flow based correlation (Jamiolahmady et al., 2006). This is due to the fact that the formulation for the f g -based correlation was obtained by momentum balance, which is not going to be affected by the choice of variable. Therefore, for f g we can write k rgm iner = 0 k 1+βρ m k μ m 1 A; ð8þ juj sm where refers to the absolute value of the quantity, β is the (singlephase) Forchheimer coefficient sometimes referred to as the non- Darcy or inertial coefficient and subscript m refers to the miscible case. In Eq. (8) the required miscible fluid properties (i.e. density, ρ m, and viscosity, μ m ) are the arithmetic average of the fluid properties of gas and liquid at any given pressure (in the vicinity of the dew point), which is a good approximation of the average values at the dew point that are not easily available Base relative permeability for gas (k rgb ) The base relative permeability curve is the measured curve at the lowest velocity level (above which k r is a function of velocity) and the highest IFT value (below which k r is a function of IFT), (k rgb ) meas, which is then modified for the effect of inertia, (k rgb ) iner when the velocity is high. The exact definition of (k rgb ) meas will be discussed later in this section. Here a similar approach to that used for the miscible curve was adopted, i.e., defining an equivalent single phase, eliminating the need for the two-phase Forchheimer factor, β g.in other words, the inertial pressure drop was calculated by using the single-phase inertial factor, β, the total velocity, u T, and the summation of total momentum inflow of gas and condensate. Therefore, k rgb that is modified for the effect of inertia is calculated by 0 1 iner k rgb k = B meas C A: ð9þ 1+ βρ avekðk rgb Þ meas juj T f g μ g In Eq. (9), the weighted average density value is calculated at the base conditions and based on the fractional flow of the two flowing phases. Eqs. (8) and (9) correctly extend the correlation to the singlephase flow limits. That is, if (k rgm ) iner from Eq. (8) or (k rgb ) iner from Eq. (9) is substituted in Darcy law, Eq. (5), and re-arranged it gives the Forchheimer Equation (1914): h jp = μ i k u + βρjuju ð10þ g Duetothepresenceoftherockproperties(k and β) in Eqs. (8) and (9), there are miscible and base relative permeability versus k rgtr curves for each core at any velocity value. The base and miscible gas relative permeability curves calculated using Eqs. (8) and (9) for Clashach and Texas Cream cores at different velocities are shown in Fig. 1a and b, respectively. It is noted that the gas miscible curve defined by Eq. (8), at low velocities where the inertial term is insignificant, will approach the k rg =k rgtr line but at high velocities the slope of the line deviates from 45 to lower values depending on the values of velocity and single-phase inertial factor for the given fluid system.similarly,the base relative permeability curve at low velocities is close to the corresponding measured value but at high velocities it is reduced. This reduction is more pronounced, at higher k rgtr values (higher gas fractional flow rates). Eqs. (9) and (10) are also valid for S wi N0, with k eg (S wi ) and β g (S wi ) replacing k and β, respectively, assuming that immobile water acts like a part of the rock. 2.. Measured base relative permeability for gas (k rgb ) meas The base-relative-permeability curve in our previous work (Jamiolahmady et al., 2006) was given at IFT=0.85 mn m 1, with

4 M. Jamiolahmady et al. / Journal of Petroleum Science and Engineering 66 (2009) Fig. 1. Base and miscible gas relative permeability versus relative-permeability ratio at three different velocities calculated using Eqs. (8) and (9), respectively, for a) Clashach core b) Texas Cream core. the recommendation for field application that the base curve should be measured at the highest applicable value of IFT for the fluid under consideration and the lowest practicable velocity. A series of relative permeability measurements were made on a Texas Cream limestone core at IFT values higher than 0.85, as shown in Fig. 2. The objective was to investigate the effect of increasing IFT on the gas and condensate relative permeability and to identify the IFT value above which the relative permeability would not be affected by velocity. The results indicated that there was a continual reduction in relative permeability with increasing IFT if k r was expressed in terms of fractional flow rate ratio (f g ), Fig. 2a. However, if the relative permeability curves were plotted versus relative permeability ratio, the effect of viscosity, fluid property, was eliminated and the variation of k r curves measured at high IFT was demonstrated to be minimal, Fig. 2b. These experimental findings suggested that the IFT value of 0.85 mn m 1 considered for the base curve in the previous study is not a universal value and it could have a value in the range of 1 to mn m 1, Fig. 2b. The two figures also show that at a given value of IFT the low velocities in the range of 5 to 50 m d 1 have a minimal effect on relative permeability, especially at high values of f g.in Fig. 2b it can be noticed that there is a slight decrease in (k rgb ) meas, which is more pronounced at higher k rgtr values, when IFT increases from 0.85 to mnm 1. These results highlight that the effect of reduction of IFT can not be compensated by an equivalent increase in velocity. Later in the following section it will be shown that the effect of viscous and capillary forces are independently accounted for using a combination of IFT and capillary number (ratio of viscous to capillary forces). Hence, both base IFT and base capillary number are required for this correlation, which has universal parameters. After careful examination of the measured data, accuracy of the measurements and the fact that at higher IFT values and/ or lower velocities the coupling effect is diminishing for all core types, we selected a base IFT and capillary number of mn m 1 and 1E 7, respectively, for measuring the base relative permeability. This value for the base capillary number was obtained using Eq. (11) as follows: μ g u g + u c Nc = ; ð11þ σ where σ is the interfacial tension. Based on the data of Fig. 2 for core samples with measurements at IFT of 0.85 mn m 1 rather than base IFT of mn m 1, Eq. (12) was used to adjust the base curve data. h i k rg h i meas 7:07 h i IFT = =0:69 + 0:1 k rgtr ð12þ k rg meas IFT = 0: Weight function formulation (Y g ) To determine the form of the weight function Y g, Eq. (1), we used experimentally measured k rg values on selected Clashach, Berea, Texas Cream and RC2 core samples. Tables 1 4 include the basic data for the

5 102 M. Jamiolahmady et al. / Journal of Petroleum Science and Engineering 66 (2009) Fig. 2. Gas relative permeability at two velocities and three IFT values for Texas Cream core sample based on (a) gas to total fractional flow rate ratio (b) gas to total, gas plus condensate, relative permeability ratio. different sets of measurements conducted on these core samples including the range of IFT and velocity values at which the relative permeability measurements were conducted. This comprehensive data bank included over 100 relative permeability measurements. A number of measurements on a number of core samples, i.e., Clashach, Texas Cream, RC and a propped fracture, Table 5, were excluded so that Table 1 Basic test data for experiments conducted on Clashach core at different conditions. Index S wi k k(s wi ) IFT β β(s wi ) Velocity a (k rgtr ) Micpoeff (%) (md) (md) (mn m 1 ) (m 1 ) (m 1 ) range E E E E E E E E8 1.95E8 b E8 1.95E8 b E E8 b E E8 b S wi is the amount of immobile water saturation and k(s wi ) and β(s wi ) are the singlephase permeability and inertia factor, respectively, of the core sample at S wi. IFT refers to interfacial tension. a (k rgtr ) Micpoeff refers to k rgtr =k rg /(k rg + k rc ) value at which the measured k rg has been affected by the presence of micro-pores. b Calculated β(s wi ) value using, Eq. (17) of Jamiolahmady et al., (2006). the evaluation exercise conducted on the performance of the correlation included data that were not used for its development. This data bank included over 200 additional relative permeability measurements. To determine the functional form of Y g, for each k rg measured experimentally at the prevailing conditions, Y g was calculated, using the following equation: k rg k rgm iner Y g = ; ð1þ k rgb iner k rgm iner which is the re-arranged form of Eq. (1). (k rgm ) iner at any velocity level and k rgtr were calculated using Eq. (8) and (k rgb ) iner was the Table 2 Basic test data for the experiments conducted on Berea core at different conditions. Index S wi k k(s wi ) IFT β β(s wi ) Velocity (k rgtr ) Micpoeff (%) (md) (md) (mn m 1 ) (m 1 ) (m 1 ) range E E , E , 0.954, E E8 a E E8 a E E8 a E E8 a a Calculated β(s wi ) value using, Eq. (17) of Jamiolahmady et al., (2006).

6 M. Jamiolahmady et al. / Journal of Petroleum Science and Engineering 66 (2009) Table Basic test data for experiments conducted on Texas Cream at different conditions. Index S wi k k(s wi ) IFT β β(s wi ) Velocity (%) (md) (md) (mn m 1 ) (m 1 ) (m 1 ) range E E9 a E9 a 5.9E9 b E9 a 5.9E9 b a β of Texas Cream with k=9.1 md, has been used due to lack of β measurement. b Calculated β(s wi ) value using, Eq. (17) of Jamiolahmady et al., (2006). Table 4 Basic test data for experiments conducted on RC2 at different conditions. Index S wi k k(s wi ) IFT β β(s wi ) Velocity (%) (md) (md) (mn m 1 ) (m 1 ) (m 1 ) range E E E E E10 a E E10 a E E10M a a Calculated β(s wi ) value using, Eq. (17) of Jamiolahmady et al., (2006). Table 5 Basic test data for experiments conducted on a number of cores at different conditions. Index Core type S wi k k(s wi ) IFT β or β(s wi ) Velocity (%) (md) (md) (mn m 1 ) (m 1 ) range 28 Clashach E E8 a Texas Cream E9 b E9 a RC E E E Propped fracture 0 146, E a Calculated β(s wi ) value using, Eq. (17) of Jamiolahmady et al., (2006). b β of Texas Cream with k=9.1 md, has been used due to lack of β measurement. experimentally measured base k rg value at any given k rgtr modified for the effect of inertia using Eq. (9). The Y g weight function depends on the capillary number ratio, the ratio of the prevailing capillary number to its base value. Similarly to the fractional flow based correlation (Jamiolahmady et al., 2006) the capillary number defined by Eq. (14) was selected to represent the ratio of viscous and capillary forces, which proved to correlate the data more consistently. Nc = kδp /σl = kjjp j /σ ; ð14þ where ϕ is the porosity and L is the core length. It is noted that the capillary number expression, in Eq. (14), is different from that of Eq. (11) used for defining the base capillary number. However, as it will be discussed later they are related to one another using Darcy law for two-phase flow, Eq. (5). The data of Fig. is the plot of calculated Y g based on the experimentally measured relative permeability on two dry (without initial water saturation) Clashach core samples at different IFT values based on k rgtr. The data in this figure demonstrate that the bulk of data follows a consistent trend but very scattered. In this figure a group of data have been marked deviating data' with low Y g values that correspond to data points at low IFT and high k rgtr values, Fig.. This different behaviour is attributed to the contribution of a different flow mechanism because of to the presence of condensate in the micropores of the core at high k rgtr values (high gas fractional flow rates). Micro-pores have been shown to affect the flow behaviour of conventional gas oil systems. Mcdougall et. al. (2002) had to use two different pore size distributions to match the P c curves of cores with micro-pores. It was noted that for deviating data of Fig., the (k rgb ) meas values are much lower than their corresponding k rg values at the same k rgtr but at lower IFT values, Fig. 4. That is, in the presence of these micro-pores at lower IFT values, condensate flow does not impede the flow of gas, i.e., k rg is close to unity. But at higher base IFT value the condensate flow restricts the flow of gas thereby reducing its k r to very low values, which results in very low Y g values, Eq. (12). If the percentage of these small pores is very high, e.g., Texas Cream, even at base IFT, the flow of gas is not affected by the flow of condensate and the (k rgb ) meas values are close to unity and to the corresponding values at lower IFT for the same high k rgtr, Fig. 5, hence Y g will not take these low values. Furthermore, it is noted that in the latter case because of to the small difference between the two interpolation limits, i.e. (k rgb ) meas and k rgm the accuracy of the weight function, Y g, does not affect the interpolated value, k rg, significantly. For the Clashach and Berea core samples the limit of this effect is different. Last columns in Tables 1 and 2 include the corresponding k rgtr values for these data points. These data indicate that for the Clashach core, where there is a very small percentage of the micropores, the contribution is limited to very high k rgtr values, Table 1, Fig.. Y g vs. Ncr for different Clashach core tests.

7 104 M. Jamiolahmady et al. / Journal of Petroleum Science and Engineering 66 (2009) Fig. 4. Gas relative permeability (k rg ) vs. relative permeability ratio (k rgtr ) for Clashach core at three different IFT values. whilst for the Berea core, which contains a higher share of small pores, the data at lower k rgtr values have also been affected, Table 2. In the presence of connate water, in these water-wet cores, the small pores are filled with water and the effect of preferential filling by condensate is not observed. Due to the unique nature of this behaviour an extra term was added to account for this micro-pores effect as, Y g = Y g Main Y g MicPoeff ; ð15þ where (Y g ) Main expresses the main trend followed by most of the data points and (Y g ) MicPoeff term accounts for the effect of micro-pores. The mathematical expression for (Y g ) Main depends on core properties (i.e., k, ϕ and β), interfacial tension (IFT), pressure gradient and the capillary number (Nc ). Different functional forms with different coefficients and exponents were obtained to express these dependencies. During this exercise care was taken to have a simpler yet more general correlation. Hence, simpler mathematical expressions compared to those reported for the fractional flow based correlation (Jamiolahmady, 2006) were sought with realistic trends for the range not covered by the available measured data, whilst ensuring that the correlation has either universal parameters, that are applicable to all types of rocks, or parameters that could be determined from commonly available petrophysical data. After careful examination of the measured data the following (Y g ) Main formulation was obtained: Y = 1+A 1 x g Main 1+A 1 x + A 2 x ; ð16þ 2 with x = log½σ r Ncr Š A 1 = C 1 + C A C 4 C 2 + A C 4 A 2 = C 1 + C 5 AC 6 C 2 + A C 6 p A = β ffiffiffi k ; where C 1 =8, C 2 =800, C =0.20, C 4 =2.20, C 5 =0.15, C 6 =0.81, σ r refers to the ratio of base IFT (σ b ) to current IFT value (σ) and Ncr refers to the ratio of Nc at any condition to the corresponding value for the base curve at the same k rgtr. The base value of Nc at any k rgtr value can be Fig. 5. Gas relative permeability (k rg ) vs. relative permeability ratio (k rgtr ) for Texas Cream core at two different IFT values.

8 M. Jamiolahmady et al. / Journal of Petroleum Science and Engineering 66 (2009) Fig. 6. (Y g ) Main vs. σ r Ncr product for Berea core tests together with best fitted curve, Eq. (16). Fig. 7. (Y g ) Mipoeff vs. σ r Ncr product for data of Clashach and Berea core tests at high k rgtr affected by micro-pores together with the best fitted curve, Eq. (18). obtained by Eq. (17), which requires the measured k rgb, fluid properties and the corresponding base Nc of 1E 7 defined earlier, f g Ncb = / k rgb meas f g Ncb = / k rgb meas 1E 7; ð17þ Eq. (17) has been obtained using Darcy law for two-phase flow, Eq. (5), and Eq. (11). Furthermore, for both the base Ncb and (k rgb ) meas, Hermite cubic spline interpolation method is used to cover the whole range of k rgtr variation. Fig. 6 is the plot of Y g calculated from measured k rg data using Eq. (1) for Berea together with the best fitted curve calculated using Eq. (16). Fig. 7 demonstrates that for the measurements affected by micropores, the difference between Y g calculated from measured k rg data using Eq. (1) and the corresponding (Y g ) Main values obtained from Eq. (16), can be expressed by a mathematical expression as, Y = 1+C 7 x g A MicPoeff 1+C 7 x + C 8 x 2 4 A 5 ; ð18þ where x=log[σ r Ncr ], C 7 = 0.11, C 8 =0.0. In Eq. (18), A 4 and A 5 terms determine the limit and extent of (Y g ) MicPoeff and are function of k rgtr, IFT and the percentage of micro-pores in the core. Different options were considered to reflect the amount of micro-pores in the system. Finally, considering the reliability and the amount of information available, S w1μm, which is the wetting phase saturation at capillary pressure (P c ) corresponding to the radius value of one micrometer obtained from mercury porosimetry P c curve, was selected. The irreducible water saturation in the cores would fill the micro-pores and hence, the flow of gas and condensate would not be affected by these pores for the corresponding measurements for cores Table 6 The percentage of micro-pores in the cores reflected by S w1 μm. Core type k ϕ S w1 μm (md) (%) (PV) Clashach Clashach Berea Texas Cream Texas Cream a RC RC Propped fracture 146, k is the absolute permeability, ϕ is the porosity, S w1 μm is the wetting phase saturation at capillary pressure corresponding to the radius value of one micrometer obtained from mercury porosimetry P c curve. a S w1 μm of Texas Cream with k=11 md, has been used due to lack of P c measurement.

9 106 M. Jamiolahmady et al. / Journal of Petroleum Science and Engineering 66 (2009) Table 7 Average absolute deviations (AAD) and standard error of estimates (SEE) between measured and calculated values of k rg for Clashach core at different conditions. Index S wi IFT AAD%-k rg SEE-k rg AAD%-k rg SEE-k rg (%) (mn m 1 ) (1) (1) (2) (2) Ave Ave Ave All AAD% and SEE refer to average absolute percentage deviation and standard error of estimates, respectively. (1) Corresponds to k rg values obtained using (Y g ) Main, Eq. (16). (2) Corresponds to k rg values obtained using Y g, Eq. (15). with S wi N0providedthatS wi NS w1μm, which was the case for all measurements conducted here. In Eq. (18), A 4 limits the effect of micro-porosity to the corresponding k rgtr and IFT values for the cores with low and moderate percentage of micro-pores (e.g., Clashach and Berea cores, Table 6)by the following equation: 2 h i C k A6 6 rgtr 7 A 4 = 4 h i A6 5 C 9 + k rgtr " # A 6 = A 7 C 9 + C 10½ IFT rš C 11 ; ð19þ C 9 + ½IFT r Š C 11 2 h i C1 6 A 9 C 10 + C 12 S w1μm 7 A 7 = 4 h i C1 5 C 9 + S w1μm where C 9 =1E 4, C 10 =1200, C 11 =.8, C 12 =100, C 1 =2.5. In Eq. (18), A 5 restricts the application of this term to the core types with high percentage of micro-pores (e.g., Texas Cream, RC 2, Table 6) and is expressed by, 2 A 5 = 1+SC 15 4 w1μm 5; ð20þ C 14 + S C 15 w1μm where C 14 =10,000, C 15 = 7.. Error analysis of relative-permeability-ratio (k rgtr ) based correlation Table 8 Average absolute deviations (AAD) and standard error of estimates (SEE) between measured and calculated values of k rg for Berea core at different conditions. Index S wi IFT AAD%-k rg SEE-k rg AAD%-k rg SEE-k rg (%) (mn m 1 ) (1) (1) (2) (2) Ave Ave All Table 9 Average absolute deviations (AAD) and standard error of estimates (SEE) between measured and calculated values of k rg for Texas Cream core at different conditions. Index S wi IFT AAD%-k rg SEE-k rg (%) (mn m 1 ) (2) (2) Ave Ave All Table 10 Average absolute deviations (AAD) and standard error of estimates (SEE) between measured and calculated values of k rg for RC2 reservoir core at different conditions. Index S wi IFT AAD%-k rg SEE-k rg (%) (mn m 1 ) (2) (2) Ave Ave All The average absolute percent deviation (AAD%) and standard error of estimates (SEE) between measured k rg values and the corresponding calculated values by the k rgtr correlation, for data used for the development of this correlation, are found in Tables In Tables 7 and 8, the AAD% and SEE values in column four and five, respectively, labelled as (1), are those obtained using (Y g ) Main, Eq. (16), for k rg estimations. The corresponding AAD% and SEE values in column six and seven, respectively, labelled as (2), are those obtained using Y g, Eq. (15), which include the (Y g ) MicPoeff term, Eqs. (18) to (20). There are lower values of AAD% and SEE in column six and seven compared to those in column four and five, respectively, for dry Clashach and Berea cores (S wi =0), which have data points at high k rgtr affected by the presence of micro-pores. The difference might seem minimal in some cases but this is due to the fact that the number of data points affected by micro-pores is very limited compared to the whole numbers of measurements. The AAD% and SEE values in Tables 9 11 are all obtained using Y g, Eq. (15). In this exercise AAD% greater than 0% were obtained for two sets of measurements conducted on Clashach and Berea, tests indexed 8 and 17. It is expected that a systematic error could partly contribute to Table 11 Average absolute deviations (AAD) and standard error of estimates (SEE) between measured and calculated values of k rg for a number of cores not used for the development of the correlation at different conditions. Index Core type S wi IFT AAD%-k rg SEE-k rg (%) (mn m 1 ) (2) (2) 28 Clashach All Texas Cream All RC All Propped fracture

10 M. Jamiolahmady et al. / Journal of Petroleum Science and Engineering 66 (2009) Fig. 8. Gas relative permeability measured in an experiment and estimated by the correlation vs. relative permeability ratio for Berea core at two IFT and velocity values. these high error values, which was noticed for all measurements in these two tests. The AAD% a of 19%, 18%, 11% and 14% are noted for Clashach, Berea, Texas Cream and RC2, respectively, Tables The corresponding SEE values are 0.09, 0.070, and 0.072, respectively. These values are considered to be satisfactory bearing in mind that this correlation has universal parameters and accounts for the combined effect of coupling and inertia. Fig. 8 is a plot of measured and calculated k rg values for the Berea core sample at two different velocities and two IFT of 0.06 and 0.15 mn m 1. These measurements are only affected by the coupling effect, i.e., there is an the increase in k r as velocity increases and/or IFT decreases. The agreements between the measured k rg and corresponding calculated k rg, are reasonably good for the whole range of k rgtr variation. Fig. 9 is a plot of measured and calculated k rg values for the RC2 core sample at two different velocities and IFT of 0.15 mn m 1. The figure shows that the correlation correctly captures the competition between coupling and inertia as demonstrated by the experimental measurements. At high k rgtr values (high gas fractional flow rates) the relative permeability values decrease when the flow rate increases, because of the dominance of the inertial effect but the positive coupling effect surpasses the inertial effect at lower k rgtr values (lower gas fractional flow rate) resulting in a net increase of relative permeability with velocity. In a separate exercise the standard error of estimates for the six fitting parameters (i.e. C 1 to C 6 ) of Eq. (16) was estimated as 6.55 E-5, 1.12E-5, 1.28E-, 2.1E-2, 4.12E-,.28E-, respectively. These SEE are much smaller than the magnitude of the parameters, which confirm that there is a decent estimate for the fitting parameters. In this exercise the 140 k r data points measured on RC2 core were used and the procedure described by Harris (1998) for Microsoft Excel Solver was followed. The performance of the correlation was then evaluated against measured data on Clashach sandstone (k=140 md), Texas Cream limestone (k=7.4 md), reservoir sample RC (k=.9 md) cores and a propped fracture medium (k=146,000 md). These measurements had not been used for the development of the correlation. The AAD% and SEE between the measured and corresponding predicted values for these data points are found in Table 11. It predicted the measured data of these cores with AAD% of 20%, 12%, 14% and 12%, respectively. The corresponding SEE values are 0.049, 0.028, 0.09 and 0.029, respectively. These reasonably low deviations confirm the integrity of the formulation, especially considering that RC core and propped fracture have significantly different core characteristics compared to the cores used in the development of the correlation, Tables It is Fig. 9. Gas relative permeability measured in an experiment and estimated by the correlation vs. relative permeability ratio for Texas Cream core at two velocity values.

11 108 M. Jamiolahmady et al. / Journal of Petroleum Science and Engineering 66 (2009) Fig. 10. Gas relative permeability measured in an experiment and estimated by the correlation vs. relative permeability ratio for Clashach core with 25% initial water saturation at two velocity values. important to note that in this exercise k, ϕ, β and S w1μm, (obtained from mercury porosimetry P c curve) and one set of (k rgb ) meas were used to predict relative permeability at different velocity and IFT values. As mentioned previously the correlations available in the literature estimate the two effects separately and include a number of parameters, which should be determined by k r measurements at low IFT and high velocity conditions. This new correlation eliminates the need for measurements of gas/condensate relative permeability at simulated near wellbore conditions, which are very demanding and expensive. Figs. 10 and 11 show that the predicted values by the correlation for a Clashach sandstone core and carbonate Texas Cream core both with 25% initial water saturation are consistent with the corresponding measured values. That is, both the measured and predicted k rg values demonstrate an improvement in relative permeability with velocity albeit to a different extent for these two cores. In the case of low permeability RC core, Fig. 12, the predicted k rg values by the correlation, similarly to the corresponding measured values, show that the relative permeability decreases due to the impact of inertia at higher k rgtr values (higher gas fractional flow rates). However at lower k rgtr values (lower gas fractional flow rates) there is an improvement in k rg with an increase in velocity, Fig. 12. In the case of a propped fracture with permeability of 146 D, k rg values are reducing with an increase in velocity due to the dominant effect of inertia at all k rgtr values, Fig. 1. The velocity range worked in this test is well beyond the range of velocity values used in the previous core tests, Tables 1 5. There is also a good agreement between the measured values and corresponding predicted values by the correlation for these results. These consistent trends have been achieved mainly because of the fact that the new correlation expresses the combined effect of coupling and inertia when the two upper (k rgm ) and lower (k rgb ) limits of correlation are modified for the effect of inertia before interpolation is carried out. As mentioned previously, the base-relative-permeability curve in our previous work (Jamiolahmady et al., 2006) was given at IFT=0.85 mn m 1, with the recommendation for field application that the base curve should be measured at the highest applicable value of IFT for the fluid under consideration and the lowest practicable velocity. In Section 2. it was discussed that this definition is not the Fig. 11. Gas relative permeability measured in the core experiments and estimated by the correlation vs. relative permeability ratio for Texas Cream core with 25% initial water saturation at two velocity values.

12 M. Jamiolahmady et al. / Journal of Petroleum Science and Engineering 66 (2009) Fig. 12. Gas relative permeability measured in an experiment and estimated by the correlation vs. relative permeability ratio for RC core at two velocity values. correct definition of the base relative permeability curve. That is, it does not satisfy the conditions that should reflect that for IFT higher than base IFT and velocities lower than base velocity, (k rgb ) meas of gas condensate systems should no longer be a function of IFT and velocity switching the conventional gas oil systems. A sensitivity study was also conducted to demonstrate the advantage of using the new relative permeability ratio based correlation presented in this manuscript compared to our previous fractional flow based correlation (Jamiolahmady et al., 2006). In this exercise two different conditions at which (k rgb ) meas had been measured, were considered. Based on these information, the predicted k rg values using these two approaches for some of the measured data on Texas Cream were then compared. The measured data correspond to the tests indexed 18 and 19 in this paper and 21 and 22 in our previous publication. The two base measurements considered in this study correspond to data of Fig. 2 measured at a) IFT of 9.4 mn m 1 and velocity of 5.2 md 1 and b) IFT of 0.85 mn m 1 and velocity of 6.1 md 1. In the case of the previous correlation two different set of (k rgb ) meas, Fig. 2a, and Ncb, Eq. (18) of Jamiolahmady 2006, were obtained. This resulted in a significant increase in AAD% between predicted and measured k rg values from a) 6% and 1% (reported in Table 5 of the previous publication) to b) 90% and 109%, respectively. In the case of the new correlation the (k rb ) meas measured at IFT of 0.85 mn m 1 were first transferred to that of the specified base IFT of mnm 1 using Eq. (12) then Ncb was calculated using Eq. (17). Fig. 2b shows that (k rb ) meas,andhencencb,remainsthesamewhen IFT is increased from to 9.4 mn m 1. Therefore, the corresponding AAD% values of 8.7 % and 11.2% reported in Table 9 is applicable for both cases. These results highlight that the use of the previous correlation is limited only to the range of data used in that study. In other words, a more general definition of both base IFT and base capillary number are required for a correlation universal parameters, as presented in this manuscript. 4. Summary and conclusions A generalised relative permeability correlation for gas condensate systems, which is based on relative permeability ratio, has been developed. It accounts for the combined effects of positive coupling and negative inertia. In this approach gas relative permeability is correlated and condensate relative permeability is obtained by the expression of relative permeability ratio. The k rg correlation interpolates between k rgb and k rgm curves, both modified for the effect of inertia, using a generalized interpolating parameter Y g, which expresses the dependency of the relative permeability to velocity and interfacial tension including micro-pore effect. The parameters of the correlation are either universal (constant for all cores) or can be obtained from commonly available petrophysical Fig. 1. Gas relative permeability measured in an experiment and estimated by the correlation vs. relative permeability ratio for a propped fracture at two velocity values.

13 110 M. Jamiolahmady et al. / Journal of Petroleum Science and Engineering 66 (2009) rock properties (i.e., permeability, porosity, single-phase inertial factor, capillary pressure curve obtained from mercury porosimetry). This eliminates the need for difficult and expensive relative permeability measurements at near well conditions. The comprehensive relative permeability data (measured in our Laboratory at Heriot-Watt University) on cores with permeability ranging from 9 md to 550 md, with the corresponding basic test data given in Tables 1 4 and 6, and different lithology (sandstone and carbonate) were used to develop the presented correlations. When the correlations were tested for a md sandstone, 7 md carbonate and 146 D propped fracture core, none of which had been used in developing the correlation, with the corresponding basic test data given in Tables 5 6, the results were very satisfactory showing a relatively low and uniformly distributed deviations. These results confirm the generality of the correlation and reliability of the information obtained from it to great extent as these measurements covered a reasonably wide range of IFT and velocity variations for porous media with very different characteristics. However, more data is being generated to further evaluate the performance of this correlation for very tight cores. That is, although the RC core sample is a relatively low permeability core but the evaluation of correlation should also be directed towards cores with permeabilities in the range of a fraction of milli-darcy. Acknowledgements The above study was conducted as a part of the Gas condensate Recovery Project at Heriot-watt University. This research project is sponsored by: The UK Department for Business, Enterprise & Regulatory Reform (BERR), BP Exploration Company (Colombia) Ltd, Eni Petroleum Co, Gaz de France, Petrobras, StatoilHydro and Total Exploration UK plc, which is gratefully acknowledged. References Ali, J.K., McGauley, P.J., Wilson, C.J., Oct The effects of high velocity flow and PVT changes near the wellbore on condensate well performance. SPE 892, Proc. of SPE Annual Technical Conference and Exhibition, Texas, pp Al-Kharusi, S.B., Jan Relative permeability of gas condensate near wellbore, and gas condensate water in bulk of reservoir. PhD thesis, Heriot-Watt University. App, J.F., Mohanty, M., Gas and condensate relative permeability at near critical conditions: capillary and Reynolds number dependence. J. Petrol. Sci. Engng. 6 (1), Asar, H., Handy, L.L., Feb Influence of interfacial tension on gas/oil relative permeability in a gas condensate system. SPE 11740, SPERE (1), Bardon, C., Longeron, D.G., Oct Influence of very low interfacial tension on relative permeability. SPEJ 20 (), Blom, S.M.P., Hagoort, J., Sept How to include the capillary number in gas condensate relative permeability functions? SPE 49268, Proc. of SPE Annual Technical Conference and Exhibition, Louisiana, pp Blom, S.M.P., Hagoort, J., Soetekouw, D.P.N., Oct Relative permeability near welbore conditions. SPE 895, Proc. of SPE Annual Technical Conference and Exhibition, Texas, pp Coats, K.H., Oct An equation of state compositional model. Soc. Pet. Eng. J. 20, Dacun, L., Thomas, W.E., May Literature review on correlations of the non-darcy coefficient, SPE Proc. of the Permian Basin oil and Gas Recovery Conference, Texas. Danesh, A., khazam, M., Henderson, G.D., Tehrani, D.H., Peden, J.M., June as condensate recovery studies. Proc. of DTI Improved Oil Recovery and Research Dissemination Seminar, London. Forchheimer, P.: Hydraulik, Chapter 15, Leipzik and Berlin, 1914, pp Harris, D.C., Jan Nonlinear least-square curve fitting with Microsoft Excel Solver. Journal of Chemical Education 75 (1), Henderson, G.D., Danesh, A., Tehrani, D.H., Peden, J.M., May The effect of velocity and interfacial tension on the relative permeability of gas condensate fluids in the wellbore region. J. Pet. Sci. Eng. 17, also in Proc. of 8th IOR Symposium Vienna, 1995, Henderson, G.D., Danesh, A., Tehrani, D.H., Peden, J.M., June Measurement and correlation of gas condensate relative permeability by the steady-state method. SPEJ SPE Henderson, G.D., Danesh, A., Tehrani, D.H., Effect of positive rate sensitivity and inertia on gas condensate relative permeability at high velocity. Pet. Geosci. 7, Jamiolahmady, M., Danesh, A., Tehrani, D.H., Duncan, D.B., Oct A mechanistic model of gas condensate flow in pores. Transp. Porous Media 41 (1), Jamiolahmady, M., Danesh, A., Tehrani, D.H., Duncan, D.B., 200. Positive effect of flow velocity on gas-condensate relative permeability: network modelling and comparison with experimental results. Transp. Porous Media 52 (2), Jamiolahmady, M., Danesh, A., Henderson, G.D., Tehrani, D.H., Dec Variations of gas condensate relative permeability with production rate at near wellbore conditions: a general correlation. SPE Reserv. Eng. Evalu. J. 9 (6), SPE 8960, also in Proc. ofthe SPE OEC, Aberdeen, UK, Sept Mott, R., Cable, A., Spearing, M., Oct Measurements and simulation of inertial and high capillary number flow phenomena in gas condensate relative permeability, SPE Proc. of SPE Annual Technical Conference and Exhibition Texas. Narayanaswamy, G., Sharma, M.M., Pope, G.A., June Effect of heterogeneity on the non-darcy flow coefficient. SPE Reservoir Eval. Eng. 2 (), Pope, G.A., Narayanaswamy, G., Delshad, M., Sharma, M., Wang, P., Sept Modelling relative permeability effects in gas condensate reservoirs. SPE Proc. of SPE Annual Technical Conference and Exhibition, Louisiana, pp Mcdougall, S.R., Cruickshank, J., Sorbie, K.S., Anchoring methodologies for porescale network models: application to relative permeability and capillary pressure prediction, Petrophysics Houston, VOL 4; PART 4, pp Whitson, C.H., Fevang, O., Saevareid, A., Oct Gas condensate relative permeability for well calculations, SPE Proc. of SPE Annual Technical Conference and Exhibition Texas.