The correlation of the steady-state gas/water relative permeabilities of porous media with gas and water capillary numbers

. The steady-state gas, k rg , and water, k rw , relative permeabilities are measured with experiments of the simultaneous ﬂ ow, at varying ﬂ ow rates, of nitrogen and brine (aqueous solution of NaCl brine) on a homo-geneous sand column. Two differential pressure transducers are used to measure the pressure drop across each phase, and six ring electrodes are used to measure the electrical resistance across ﬁ ve segments of the sand column. The electrical resistances are converted to water saturations with the aid of the Archie equation for resistivity index. Both k rw and k rg are regarded as power functions of water, Ca w , and gas, Ca g , capillary numbers, the exponents of which are estimated with non-linear ﬁ tting to the experimental datasets. An analogous power law is used to express water saturation as a function of Ca w , and Ca g . In agreement to earlier studies, it seems that the two-phase ﬂ ow regime is dominated by connected pathway ﬂ ow and disconnected ganglia dynamics for the wetting ﬂ uid (brine), and only disconnected ganglia dynamics for the non-wetting ﬂ uid (gas). The water saturation is insensitive to changes of water and gas capillary numbers. Each relative permeability is affected by both water and gas capillary numbers, with the water relative permeability being a strong function of water capillary number and gas relative permeability depending strongly on the gas capillary number. The slope of the water relative permeability curve for a gas/water system is much higher than that of an oil/water system, and the slope of the gas relative permeability is lower than that of an oil/water system.


Introduction
The relative permeability functions of reservoir rocks are fed as input data in macroscopic simulators of enhanced oil and gas recovery processes, and affect the numerical predictions of the spatio-temporal evolution of hydrocarbon saturation, and implicitly, the design of strategies for reservoir exploitation (Kamali and Hussain, 2017;Sherafati and Jessen, 2017).
The relative permeability functions of porous rocks for a specific fluid system (gas/water, gas/oil, oil/water, gas/oil/ water) are commonly measured with special core analysis tests performed on representative rock samples under transient (Sidiq et al., 2017) or steady-state (Reynolds and Krevor, 2015) conditions. Alternatively, computational methods such as the Lattice-Boltzman technique (Ramstad et al., 2012;Zhang et al., 2016), the pore network modeling (Ramstad and Hansen, 2006;Grøva and Hansen, 2011), and the tube bundle model (Wu et al., 2016) might be used to calculate the relative permeabilities from information concerning the pore space morphology, wettability, and fluid dynamics. The steady-state two-phase flow is established by injecting two fluids through a porous medium at fixed flow rates, and has extensively be analyzed by using hierarchical simulations at pore-and network-scale (Constantinides and Payatakes, 1996;Ramstad and Hansen, 2006;Sinha and Hansen, 2012;Valavanides, 2012;Valavanides et al., 2016), and systematic experimental approaches in model porous media (Avraam and Payatakes, 1995a, 1995b, 1999Tsakiroglou et al., 2007;Gutierrez et al., 2008;Tallakstad et al., 2009aTallakstad et al., , 2009bErpelding et al., 2013;Tsakiroglou et al., 2015). One of the main conclusions is that the wetting fluid (water) retains its connectivity along its flow path while the nonwetting fluid (oil or gas) may move either as a connected pathway or as a population of disconnected ganglia which undergo dynamic breakup and coalescence (Avraam and Payatakes, 1995a;Valavanides et al., 1998;Tallakstad et al., 2009aTallakstad et al., , 2009b. Ganglion dynamics may contribute to the understanding of capillary trapping for supercritical CO 2 under reservoir conditions (Wang et al., 2013).
In the present work, experiments of the steady-state flow of nitrogen/brine (NaCI solution) through a sand column equipped with electrodes are used to determine the gas/water relative permeabilities, and water saturation, and quantify their correlations with the current flow rates, in terms of power laws. Such information might be helpful for a variety of practical applications: the fate of CO 2 ganglia during its storage in deep aquifers or depleted oil reservoirs (Terzi et al., 2014); the enhanced oil recovery by foams (Eftekhari and Farajzadeh, 2017) or Water-Alternating-Gas (WAG) (Sherafati and Jessen, 2017); the exploitation of methane hydrates (Johnson et al., 2011). 2 Materials and methodology

Experimental setup
Experiments of the steady-state and simultaneous flow of an aqueous and a gas phase were performed on a long horizontal column of PVC which was filled with well-sorted sand (Tab. 1) and equipped with six ring electrodes and an in-house constructed multi-point resistivity meter for the measurement of the electrical conductivity across various segments of the column ( Fig. 1; Aggelopoulos and Tsakiroglou, 2008). All measuring devices were connected with the data acquisition card of a host computer. The apparatus was placed inside a thermostatted chamber to keep the temperature constant and equal to 25°C and avoid any temperature-induced disturbances of the electrical conductance. The multiple-point resistivity meter allowed the simultaneous measurement of the electrical resistance across various segments of the column with respect to a reference electrode. The total pressure drop across each phase was measured with pressure transducers connected to the inlet tubes (Fig. 2).
Initially, the sand column was evacuated and filled with brine (NaCl aqueous solution of concentration, C NaCl = 20 g/L) by capillary imbibition. Then, gas nitrogen was injected at constant flow rate, q g = 10 mL/min, with the aid of a gas mass flow controller, until reaching an irreducible wetting phase saturation. Afterwards, simultaneously with N 2 , brine started to be injected at a flow rate 0.2 mL/min (Fig. 1). Steady-state conditions were achieved when the time-averaged values of gas and brine inlet pressures along with the measured electrical conductances were stabilized. The gas flow rate was kept constant and the brine flow rate was increased stepwise from 0.2 to 5 mL/min by successive steps (Fig. 3). At each step, the steady-state space-averaged relative permeabilities were calculated for the gas, hk rg i, and aqueous, hk rw i, phase by using the Darcy law and the time-averaged pressure drop across each phase, respectively where u w , u g are the water and gas superficial velocities; A, L are the cross-sectional area and length of the porous medium; l w , l g are the water and gas viscosities; DP w , DP g are the pressure drops across the aqueous and gas phase; P m = (P g,in + P g,out )/2 is the mean gas pressure and P g,in , P g,out are the inlet and outlet gas pressures.
Given that the range of gas pressures is quite low (gauge pressures~1-5 kPa), the effect of gas compressibility on measured absolute and relative permeability is negligible and the last term of the right-hand side of equation (1b) was ignored. The brine flow rate was kept constant and the gas flow rate was reduced to q g = 5.0 mL/min (Fig. 3). When steady-state two-phase flow conditions were established, the gas flow rate was kept constant and the brine flow rate started to decrease stepwise from 5.0 to 0.5 mL/min, so that, at each step, the steady-state relative permeability was determined for each phase (Fig. 3). The entire procedure was repeated for gas flow rates q g = 2.0, 1.0 and 0.5 mL/min (Fig. 3). Totally, the gas and water relative permeability along with the water saturation were determined for 25 steady-states, associated with 25 pairs of gas and brine flow rates.
At the end of each steady-state step, the total mass of brine, collected in a storage tank connected to the outlet port ( Fig. 2), was weighted and the water saturation averaged over the column, hS w i, was estimated. The electrical resistivity index, I R , of a porous medium partially saturated with an electrolyte solution is defined as the ratio of the measured electrical resistance to the resistance that would have been measured if the porous medium had fully been saturated with the electrolyte solution of equal concentration. The measured, I R À hS w i, datasets were fitted to the Archie law to estimate the saturation exponent, n. The values of resistivity index, (I R ) ij , measured across each segment of the column were substituted in equation (2) to calculate the water saturation, hS w i ij , averaged over each column segment. It should be mentioned that the tests k 25 Â 10 À12 m 2 Aqueous phase viscosity l w 0.95 Â 10 À3 Pa s Gas phase viscosity l w 1.8 Â 10 À5 Pa s Viscosity ratio j~0.02 Interfacial tension c wg 72 Â 10 À3 N m À1 * It is considered equal to the mean value of the lower (~125 lm) and upper (~250 lm) limit of grain size range. ** It is considered equal to the diameter of the circle that is inscribed in the "window" left between three circles of equal diameter d g and touching each other (d p = 0.15 d g ).
of the simultaneous two-phase flow started from the initial condition of irreducible wetting phase saturation (drainage) of the porous medium by varying the flow rates in the stepwise manner shown in Figure 3. It is not clear if and how the saturation history, namely the succession of the fluid injection steps, might affect the various flow regimes and measured relative permeability curves. On the other hand, if the tests started from the initial condition of residual non-wetting (gas) phase saturation (imbibition), then it would be expected that the different spatial distribution of fluids might result in different flow regimes and relative permeability curves for the same capillary numbers. Notwithstanding, a systematic parametric analysis is required for addressing the aforementioned issues.

Relative permeabilities and numerical fitting
Usually, the measured relative permeabilities are expressed as functions of saturation by using Corey-type models, albeit they depend on a variety of parameters. Given that the same water saturation might be reached at two different pairs of flow-rates for the gas/water system, the corresponding relative permeabilities should be fitted with Corey-type models with the fitting parameters being functions of the flow rates (Tsakiroglou et al., 2015).
To avoid the abovementioned contradiction, we introduce a correlation where the water saturation and relative permeabilities are regarded as dependent variables. For each experiment, the water saturation, hS w i, along with the gas, hk rg i, and water, hk rw i, relative permeability, measured as space-averaged variables across the entire column, are considered as functions of two independent variables: the capillary numbers of aqueous, Ca w , and gas, Ca g , phases, defined by where l i = viscosity of phase i (i = w, g); q i = flow rate of phase i (i = w, g); c gw = interfacial tension; A = cross-sectional area of porous medium. The dependence of hk rw i, hk rg i, hS w i on Ca w , Ca g , is described by power laws of the form (Tsakiroglou et al., 2015) k rw h i ¼ a w Ca w bw Ca g bg ; ð4Þ Conductivity meter 3 Results and discussion

Resistivity index and water saturation profiles
Initially, the Archie law, equation (2), was used to correlate the steady-state electrical resistivity across the entire column, I R16 , with the total water saturation, S w16 , calculated from mass balance of effluents. A saturation exponent n = 2.01 with very satisfactory correlation coefficient was obtained (Fig. 4a). It is assumed that the steady-state values of resistivity index are also applicable to the transitional periods between successive steps, during which momentum transfer and immiscible displacement may take place (Tsakiroglou et al., 2015). Accounting for the negligible viscous resistance of the gas phase, this assumption is reasonable. Regarding the uncertainty of S w measurements from I R in accordance with Archie law, it should be reported that a variation in the value of resistivity index or saturation exponent~5% is reflected in a variation in the value of water saturation~2.5% or 3.5%, respectively. From this point of view, the error embedded in the values of water saturation resulting from Archie law is acceptable. It was observed that at high Ca g values (Fig. 4b), the resistivity index across various segments of the sand column is a clearly decreasing function of Ca w (Fig. 4b), and subsequently the corresponding water saturation is an increasing function of Ca w . At low Ca g values, the resistivity index (and hence the water saturation) changes weakly with Ca w (Fig. 4c). Overall, a weak variation of water saturation with Ca w is expected.
The effect of the Bond number on the immiscible displacement pattern, resistivity index, and saturation exponent of a porous medium has been analyzed in detail with transient two-phase flow tests performed on the same sand column at varying orientations (Aggelopoulos et al., 2005). In the present work, all tests were conducted at horizontal orientation, and therefore eventual effects of Bond number on the two-phase flow regime are expected to be evident (if exist) at the scale of the column diameter, D. At this length scale, the Bond number is approximated by the relation where Dq is the density difference of the two fluids (brine and gas) and g is the acceleration of gravity. Supposing that Dq = 1000 kg/m 3 and g = 9.81 m/s 2 and using the data of Table 1, we get Bo = 0.048, and therefore the effects of gravity forces on the two-phase flow regime, over the scale of the column diameter, can be neglected. However, if the sand column was placed in vertical position, then at the scale of column length, L, the Bo~0.3, and gravity may start affecting the two-phase flow regime. For any step change from a pair to another pair of flow rates, there is a transitional period during which the measured variables (pressures, resistivity index) change (Figs. 5a-5d). Steady-state is established when the measured fluid pressures are stabilized (Figs. 5a and 5c) and the resistivity index measured over the various column segments tends to be stabilized (Figs. 5b and 5d). In general, the period of time required to establish steady-state conditions decreases with the flow rates increasing, and on average, this period was~10 min (Fig. 5).
The exponents b w , b g , e w , e g , c w , c g , quantify the dependence of hk rw i, hk rg i, hS w i on the flow rates, viscosity ratio, porous medium properties, and wettability. For oil/water systems where the non-wetting to wetting fluid viscosity ratio, j, ranges from 0.65 to 3.35 (Avraam and Payatakes, 1995a, 1995b, 1999Tsakiroglou et al., 2007Tsakiroglou et al., , 2015 the flow regime of Connected Pathway Flows (CPF) in oil phase becomes evident at low water saturations (associated with low Ca w values), high oil flow rates (namely high, Ca o , values) and relatively high values of viscosity ratio, j Payatakes, 1995a, 1995b). In contrast, at higher water saturations (or high Ca w values), and lower values of j, the non-wetting fluid is mainly disconnected into ganglia moving and undergoing coalescence, break-up, fission and trapping, so that the flow regime is dominated exclusively by various types of oil Ganglion Dynamics (GD): Large Ganglion Dynamics (LGD), Small Ganglion Dynamics (SGD), Drop Traffic Flow (DTF) (Avraam and Payatakes, 1995a). In the present case of a gas/water system with a low viscosity ratio, j = l g /l w = 0.02, and high values of water saturation, even at relatively low Ca w values (Fig. 6), no connected pathway flow of gas phase is expected, unless the gas flow rate becomes so high that the Darcy's law ceases. Therefore, the gas phase flow regime is expected to be dominated by GD over the full range of Ca w and Ca g values, examined.
With the aid of mass balances, values of the resistivity index measured over cumulative segments of the column (Figs. 4b and 4c) were converted to water saturation values, hS w i ij (ij = 12, 23, 34, 45, 56), averaged over successive segments of the column. These saturations are plotted as function of the dimensionless axial distance of the middle of each segment from the column inlet, (n m ) ij (n m = x m /L) (Fig. 6).
Evidently the water saturation averaged over each of the five successive segments of the column, hS w i ij (ij = 12, 23, 34, 45, 56) varies noticeably (Fig. 6) and this might be attributed to the non-fully developed two-phase flow at the vicinity of inlet and outlet ports, and the capillary end effect (Aggelopoulos and Tsakiroglou, 2008). When both fluids are injected simultaneously, the irreducible water saturation, trapped in the porous medium at the end of displacement, reconnects with the bulk phase so that water saturation increases rapidly in the first segments of the column (Fig. 6a).

Dependence of relative permeabilities on capillary numbers
With the aid of Athena Visual Studio (Stewart and Caracotsios, 2008), the experimental results were fitted to equations (4)-(6) to estimate all unknown parameters (Tabs. 2 and 3). Assuming that the water saturation averaged over each segment follows the power law  then, for fixed values of Ca w and Ca g , the observed local variation of hS w i ij across the column (Fig. 6) is reflected in the variation of the pre-exponential coefficient (a s ) ij from segment to segment and might be attributed to the different boundary conditions prevailing at the ends of each segment. From the mass balance of water saturation, the column-averaged coefficient, a s , is expected to be equal to the arithmetic average of local coefficients, (a s ) ij , namely The experimentally measured and numerically estimated, equations (4)-(6), column-averaged relative permeability of aqueous, hk rw i, and gas hk rg i phase along with the column-averaged water saturation are shown as functions of capillary numbers in Figure 7, and the estimated parameter values are given in Tables 2-4. The water saturation datasets were fitted to equations (6) and (8) to estimate the parameters c w , c o , (a s ) ij ij = 12, 23, 34, 45, 56 and calculate the parameter a s , from equation (9) (Tab. 2).
In general, over the range of experimentally measured water saturations, hS w i (~0.6-0.8), hk rw i is low (Fig. 7) since the compact structure of wetting fluid is fragmented into moving or trapped ganglia of gas phase (bubbles), so that a significant fraction of water saturation belongs to "islands" which are immobile or move slowly and their contribution to hk rw i is nil. The relative permeability of aqueous phase is more sensitive to Ca w and less sensitive to Ca g (Fig. 7,  Tab. 3). As Ca w increases, the aqueous phase starts occupying a gradually increasing number of relatively narrow but non-interrupting paths spanning the porous medium and "transferring" most of the flow (Tallakstad et al., 2009a) so that the hydraulic conductivity of wetting fluid or equivalently the value of hk rw i increases appreciably (Fig. 7). The viscous coupling between gas and aqueous phase is reflected in the weak but negative effect of Ca w on hk rg i (Fig. 7, Tab. 4). As mentioned earlier, the flow regime of gas phase is governed by GD, the increase of Ca w is associated with fewer ganglia of gas phase with smaller mean and cut-off sizes (Tallakstad et al., 2009a), so that a weak decrease of hk rg i is observed (Avraam and Payatakes, 1995a;Tsakiroglou et al., 2015).  5. Transient responses of the (a, c) water/gas inlet pressures, (b, d) resistivity index across cumulative segments of the sand column for (a, b) q w = 0.2 mL/min, q g = 10 mL/min, and (c, d) q w = 1.0 mL/min, q g = 2.5 mL/min. Fig. 6. Variation of the axial distribution of column segment-averaged water saturation as a function of capillary number: (a) Ca g = 2.82 Â 10 À8 ; (b) Ca g = 1.41 Â 10 À8 ; (c) Ca g = 5.65 Â 10 À9 ; (d) Ca g = 2.82 Â 10 À9 ; (e) Ca g = 1.41 Â 10 À9 . Likewise, the gas relative permeability is more sensitive to Ca g and less sensitive to Ca w (Fig. 7, Tab. 4). As Ca g increases, a progressively higher number of pore clusters occupied by trapped gas phase ganglia is mobilized, leading to a higher number of moving ganglia and resulting in a significant increase of hk rg i (Fig. 7). At increasing Ca g values, the size of ganglia is expected to decrease (Avraam and Payatakes, 1995a). On the other hand, the hk rw i is affected weakly by Ca g since water retains its hydraulic continuity over the full range of Ca g values, through connected pathways and thin films or pendular rings surrounding the solid grains. The increased population of gas-phase ganglia, caused by the increase of Ca g leads to a relatively limited reduction of the size of "red" bonds, with the highest contribution fraction to the wetting fluid flow, and therefore to a weak reduction of hk rw i (Fig. 7).
The results of the present study along with corresponding ones from earlier studies concerning oil/water fluid systems tested in a 3-D sand column (Tsakiroglou et al., 2015) and a 2-D glass-etched pore network (Avraam and Payatakes, 1995a) are summarized in Table 5, in terms of the viscosity ratio of non-wetting (nw = o, g) to wetting (w) fluid. It is worth mentioning that: (a) each relative permeability is affected by both Ca w and Ca nw (Tab. 5); (b) the exponents b w and e nw are higher in a 3-D rather than a 2-D pore system; (c) due to the small hydraulic resistance of non-wetting phase (gas), the dependence of water saturation on Ca w and Ca nw becomes very weak (namely the values of c w , c nw tend to zero), for a 3-D system and low viscosity ratio (Tab. 5); (d) regarding the non-wetting fluid relative permeability (Fig. 8), as the viscosity ratio decreases, the wetting fluid is facilitated to occupy connected flow pathways, the fraction of non-wetting fluid belonging to trapped ganglia is enhanced, and the slope, k nw , decreases, for both 3-D and 2-D pore systems (Tab. 5); (e) particularly with reference to the wetting phase relative permeability (Fig. 8), the slope, k w , for the gas/water system is much higher than the corresponding one estimated from tests of an oil/water system (viscosity ratio j = 1.0) on the same porous medium (Tab. 5). The low hydraulic conductance of the wetting fluid is due mainly to the presence of abundant trapped ganglia of gas (bubbles) or oil in the pore network. In a gas/water system, with the Ca w increasing, due to the negligible viscous resistance of gas phase, the mobilization of gas bubbles is facilitated even for a small increase of water saturation so that the water conductance might increase respectably. In contrast, in an oil/water system, a fraction of the momentum earned with the Ca w increasing, is dissipated in configurational changes of trapped oil phase, making more difficult the mobilization of oil ganglia, and leading to moderate increase of water conductance.

Impact on core analysis and reservoir engineering
Commonly, the steady-state relative permeabilities of reservoir rocks are measured on a core plug, with well-known characteristics of porosity, by co-injecting at varying values of flow rates, a wetting, w, and a non-wetting, nw, fluid with well-known physicochemical properties (l w , l nw , c wnw , h e ). The measured relative permeabilities are simply expressed as functions of wetting fluid saturation. However, the present work along with earlier ones (Avraam and Payatakes, 1995a, 1995b, 1999Tallakstad et al., 2009aTallakstad et al., , 2009bTsakiroglou et al., 2007Tsakiroglou et al., , 2015 reveals that both the wetting fluid, k rw , and non-wetting fluid, k rnw , relative permeability along with the wetting fluid saturation, S w , are governed by the wetting fluid, Ca w , and nonwetting fluid, Ca nw , capillary number, namely Therefore, a new formulation of the two-phase flow equations is required where k rw , k rnw , S w , are handled as dependent variables and Ca w , Ca nw , as independent variables. In addition, earlier studies have shown that the dynamic capillary pressure curve measured by rate-controlled immiscible displacement tests depends also on the flow rates (Tsakiroglou et al., 2003;Aggelopoulos and Tsakiroglou, 2008;Joekar-Niasar et al., 2010). Subsequently, a new generalized numerical approach of the transient two-phase flow in porous media might be on Ca w -, Ca nw -dependent two-phase flow functions (k rw , k rnw , S w , P c ). From this point of view, it might be necessary to mention not only the wetting fluid saturation but also the Ca w , Ca nw values, when conducting steady-state 2-phase flow tests on cores of reservoir rocks. 0.7466 ± 0.110 Table 3. Estimated values of parameters relating hk rw i with Ca w and Ca g , equation (4). All parameters are unitless.
Parameter Mean value ± standard deviation a w 13786.6 ± 32500 b w 1.01205 ± 0.1425 b g À0.144 ± 0.0584 Fig. 7. Column-averaged water saturation and water/gas relative permeability as functions of Ca w for various values of Ca g . The coloured arrows indicate the vertical axis reference (saturation or relative permeability) for the corresponding lines.

Conclusion
Experiments of the steady-state N 2 /brine flow in a sand column, for varying values of gas and brine capillary numbers, are analyzed in the framework of rate-dependent relative permeability functions. The pressure drop across each phase is measured with differential pressure transducers. The axial profile of brine saturation is measured by recording the transient response of the resistivity index across five segments of the porous medium with an in-house constructed multi-point electrical conductivity meter. The effect of the gas and water capillary numbers on both gas and water relative permeabilities is quantified by estimating the relevant exponents of power laws, and the results are summarized along with those of earlier studies for oil/water fluid systems. The main conclusions are outlined below: In agreement with earlier studies, it seems that water is flowing in the form of connected pathways and disconnected ganglia, while the gas flow regime is dominated by disconnected ganglia (bubbles). The water saturation averaged over the sand column changes weakly with the water and gas capillary numbers. Each relative permeability is affected by both water and gas capillary number, the water relative permeability is a strong function of water capillary number, and the gas relative permeability changes strongly with the gas capillary number. The slope of the water relative permeability curve for a gas/water system is much higher than that of an oil/ water system, and the slope of the gas relative permeability is lower than that of an oil/water system.   Table 4. Estimated values of parameters relating hk rg i with Ca w and Ca g , equation (5). All parameters are unitless.