energies Article
Experimental Determination of Gas Relative Permeability Considering Slippage Effect in a Tight Formation Guangfeng Liu 1, * 1
2 3
*
ID
, Zhaoqi Fan 2 , Yang Lu 3 , Siying Li 1 , Bo Feng 1 , Yu Xia 1 and Qimeng Zhao 1
CMOE Key Laboratory of Petroleum Engineering, China University of Petroleum-Beijing, Beijing 102249, China;
[email protected] (S.L.);
[email protected] (B.F.);
[email protected] (Y.X.);
[email protected] (Q.Z.) Chemical & Petroleum Engineering, University of Kansas, Lawrence, KS 66045, USA;
[email protected] School of Information Engineering, China University of Geosciences-Beijing, Beijing 100083, China;
[email protected] Correspondence:
[email protected]; Tel.: 86-10-8973-2192
Received: 1 February 2018; Accepted: 19 February 2018; Published: 23 February 2018
Abstract: In this paper, the gas relative permeability considering slippage effect has been experimentally examined under various experimental conditions (i.e., ambient, high confining pressure, and high temperature). Experimentally, Klinkenberg permeabilities of 12 core samples have been measured by using steady-state flow experiment. It has been found that the Klinkenberg permeability is independent of the experimental temperature and dramatically decreases as confining pressure is increasing. Furthermore, linear correlations have been newly developed between the Klinkenberg permeability and the gas-measured permeability under various conditions. Subsequently, the developed correlations are correspondingly applied to calibrate the gas relative permeability. It has been found that the gas relative permeability can be overestimated without consideration of the slippage effect, i.e., Klinkenberg effect. In addition, the newly developed correlations have been applied to analyze the sensitivity of gas–water relative permeability to gas-measured permeability, confining pressure, and temperature. It is demonstrated that mobile water greatly alleviates the gas relative permeability in comparison to irreducible water. Although an increased confining pressure simultaneously reduces the effective water phase and gas phase permeability, the gas relative permeability increases and the water relative permeability decreases as the confining pressure increases. It is attributed to the fact that the effective water phase permeability is more sensitive to the confining pressure. Given an elevated experimental temperature, the gas relative permeability is reduced while the water relative permeability is enhanced, implying the significance of temperature effect on gas–water relative permeability measurements. Keywords: tight formation; slippage effect; Klinkenberg permeability; gas relative permeability; pore-throat structure
1. Introduction Natural gas plays an increasingly significant role in world energy supply of modern economy since fossil fuels, e.g., petroleum fluids and coal, readily trigger on environmental issues, e.g., air pollution and excessive greenhouse gas emissions [1]. Alternatively, natural gas serves as a much cleaner energy with zero solid particle emission and lower waste gas production. The gas production from tight formations is dramatically booming due to prominent improvements in hydraulic fracturing and horizontal drilling [2,3]. It has been addressed that tight gas production has increased from 11% in 1990 to 18% in 2003 of total U.S. natural gas production while, by 2010, 24.6% of the produced
Energies 2018, 11, 467; doi:10.3390/en11020467
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natural gas originated from the tight gas formations in China [4,5]. Conceptually, the recovery of a tight gas reservoir is significantly dominated by the flow capacity of gas and water. In practice, the gas production is substantially impacted by the presence of water production, especially for the late stage of the gas reservoir development. Since the relative permeability is commonly used to represent the characteristic of pore structures, saturation distribution, and wettability of the tight formation, the gas–water relative permeability has been used to demonstrate the relative flow capacity of gas and water in tight formations. Therefore, it is of significant importance to examine the gas–water relative permeability under various conditions for better understanding the underlying mechanisms of gas flow in tight formations. The gas–water relative permeability in tight formation has attracted substantial attention in past decades [6–10]. Numerous efforts have been made to theoretically and experimentally estimate the gas-measured permeability and the gas relative permeability conditioned to slippage effect [11–16]. In the laboratory, the gas–water relative permeability can be determined by either steady-state or unsteady-state flow experiments [7,14,17]. Rushing et al. (2003) addressed that the gas relative permeability may be overestimated if the two-phase gas slippage, i.e., Klinkenberg effect, was not considered in the laboratory experiments. Moreover, the confining pressure and water saturation dramatically affect both the gas phase permeability and gas relative permeability [18,19]. In addition, Walls et al. (1982) have demonstrated that accurate acquaintance of in situ fluid saturation is crucial before gas production rates can be estimated. They also indicated that confining pressure might result in significant permeability reduction according to observations in sandstones from Spirit River formation of Alberta, Canada [20]. Furthermore, it was indicated that the Klinkenberg-corrected permeability, thereafter called absolute permeability in this paper, is independent of temperature, although the Klinkenberg slippage factor is linearly proportional to the experimental temperature [21,22]. The quantification of the slippage effect on gas-measured permeability is a time-consuming process, especially when high pressure and escalated temperature are involved. The unsteady-state pressure-fall-off technique has been applied to greatly reduce the measurement duration by using toluene as wetting phase and nitrogen as non-wetting phase, respectively [10]. The slippage effect can be moderately reduced by exerting a back-pressure at the outlet of the core samples [16]. It was also demonstrated that the slippage effect may be eliminated while the back-pressure rise to a specific back-pressure and thus the non-slippage gas-measured permeability can be experimentally determined. However, the high pressure may be involved in the experimental apparatus while the core samples are sufficiently tight, resulting in more difficulties of conducting displacement experiments. As such, it is imperative to find a method of efficiently and accurately determining the gas–water relative permeability conditioned to slippage effect. In this paper, the gas–water relative permeability considering slippage effect has been experimentally examined under various experimental conditions (i.e., ambient, high confining pressure, and high temperature). More specifically, steady-state flow experiments have been conducted to establish correlations between the Klinkenberg permeability and gas-measured permeability under the aforementioned experimental conditions. Subsequently, the developed correlations are applied to correct the effective gas phase permeability for calibrating the gas–water relative permeability. In addition, the sensitivity of calibrated gas–water relative permeability to gas-measured permeability, confining pressure, and temperature have also been demonstrated by use of the unsteady displacement experiments for the purpose of reducing experimental time. 2. Experimental 2.1. Core Samples Preparation Core samples have been collected from two adjacent wells in a tight gas reservoir located in the Ordos Basin, China. The depth of payzone is ranging from 3200 m to 3400 m. The original fluid pressure of the reservoir is approximately 25.00 MPa, implying that it is a low-pressure reservoir.
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The reservoir temperature ranges from 90 ◦ C to 110 ◦ C. To simulate reservoir conditions in the laboratory, the experimental temperature is set to be between 25 ◦ C to 90 ◦ C while the experimental confining pressure is ranging from 2.35 MPa to 50.00 MPa. The gas-measured permeability, porosity, and geometrical parameters of 17 core samples used in following experiments have been listed in Table 1. Diameters of those core samples are varying from 2.508 cm to 2.521 cm, while their lengths are between 6.668 cm and 7.452 cm. Geometrical similarities of the core samples are anticipated to reduce the probable uncertainties with respect to scale differences. It is worth pointing out that the gas-measured permeability in Table 1, i.e., k∗g , was measured by using gas (i.e., nitrogen) under laboratory conditions. Table 1. Properties of 17 core samples collected from the tight gas reservoir Group No.
Core No.
kg∗ , mD
Porosity, %
Length, cm
Diameter, cm
Group #1
#1 #2 #3 #4 #5 #6 #7 #8 #9 #10 #11 #12
0.0535 0.3427 0.2046 0.4264 0.1371 0.4327 0.0952 0.1464 0.3010 0.0511 0.0929 0.3058
7.12 12.39 7.81 10.97 8.35 9.21 4.45 8.10 9.81 5.85 7.36 9.89
6.942 7.215 6.872 6.928 6.927 7.237 7.452 6.844 6.738 7.328 6.924 6.668
2.511 2.517 2.515 2.509 2.512 2.513 2.516 2.514 2.512 2.521 2.512 2.514
Group #2
#13 #14 #15 #16 #17
0.1418 0.2986 0.7312 2.8730 0.2280
7.06 7.97 8.18 13.61 5.79
7.076 6.786 7.452 7.084 6.766
2.514 2.508 2.516 2.516 2.512
Regarding the porosity, its measurement follows 5 steps: (1) Dry the core samples under 110 ◦ C for 8 h aiming to deprive connate water of the core samples for improving the accuracy of porosity measurement; (2) Weight the dried core samples; (3) Vacuum the core samples; (4) Saturate the vacuumed core samples with synthetic brine and weight the wet core samples; (5) Calculate the porosity by use of weight difference between the dry and wet core samples. Note that 17 core samples have been divided into two groups. Group #1 including 12 core samples was used to quantify the Klinkenberg permeability under ambient condition, high-temperature (HT) condition, and high-temperature-high-confining-pressure (HTHCP) condition. As for the ambient condition, the confining pressure (pcp ) is approximately 2.0 MPa higher than the mean pressure over the core sample (see GB/T 29172-2012). Given various mean pressure, the measured gas permeability changes as well. Group #2 including 5 core samples was utilized to evaluate sensitivities of calibrated gas–water relative permeability to k g , confining pressure, and temperature with the consideration of the slippage effect. 2.2. Fluid Preparation Synthetic brine with a concentration of 8.0 wt % has been prepared by dissolving potassium chloride (KCl) into deionized water to reach the same salinity as the formation water. This was to reduce the probability of permeability reduction resulted from reactions between injected water and the reservoir rock. In addition, the nitrogen of 99.999% purity has been used as a gas phase in displacement experiments. 2.3. Apparatus The apparatus has been illustrated in an experimental flowchart as shown in Figure 1. A temperature control system with a range of 25.0 ◦ C to 200.0 ◦ C and an accuracy of ±0.2 ◦ C has been used to control the experimental temperature (see (17) in Figure 1). As for high-temperature
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experiments, a tubing with a length of 4.0 m and an inner diameter of 0.003 m has been placed in the temperature control system for transporting and heating the gas (i.e., nitrogen). The rest includes a core EnergiesEnergies 2018, 11, x FOR REVIEW holder, cylinders (capacity of PEER 100 mL, service pressure of 50.00 MPa), syringe pumps4 of(100DX, 2018, 11, x PEER FOR REVIEW 14 4 of 14 Teledyne Isco Inc., Lincoln, NE, USA), pressure transducers (maximum of 69,000.00 kPa; accuracy of ±0.01 kPa), experiments, a tubing a length 4.0mmand and an an inner of of 0.003 m has beenbeen placed in the in the experiments, a tubing withwith a length ofof4.0 innerdiameter diameter 0.003 m has placed gas mass flow controller (0–200 sccm with the accuracy of ± 0.2%F.S, Porter, Hatfield, PA, temperature control system transportingand and heating heating the (i.e., nitrogen). The The rest includes a temperature control system forfor transporting thegas gas (i.e., nitrogen). rest includes aUSA), gas core holder, cylinders (capacity of 100 mL, service pressure of 50.00 MPa), syringe pumps (100DX, mass flowcore meters (50cylinders sccm with the of accuracy of ± (0.8%Rdg 0.2%F.S), ALICAT, Tucson, holder, (capacity 100 mL, service pressure of + 50.00 MPa), syringe pumps (100DX,AZ, USA), Teledyne Isco Inc., Lincoln, NE, USA), pressure transducers (maximum of 69,000.00 kPa; accuracy of Teledyne Isco Inc., Lincoln, NE, USA), pressure transducers (maximum of 69,000.00 kPa; accuracy of and a high-pressure and high-temperature viscometer (VISCOlab PVT, CVI, Boston, ±0.01 kPa), gas mass flow controller (0–200 sccm with the accuracy of ±0.2%F.S, Porter, Hatfield, PA,MA, USA) with ±0.01 kPa), gas mass flowmeters controller (0–200 sccm with the accuracy of+ ±0.2%F.S, Porter, Tucson, Hatfield, PA, USA), gas mass flow (50 sccm with the accuracy of ± (0.8%Rdg 0.2%F.S), ALICAT, a measurable range of 0.02–10,000.00 cP and an accuracy of ±1.0% FS. The allowable pressure and USA), AZ, gas USA), mass and flowa high-pressure meters (50 sccm with the accuracy of ± (0.8%Rdg + 0.2%F.S), Tucson, and high-temperature viscometer (VISCOlab PVT, CVI,ALICAT, Boston, MA, ◦ C, respectively. temperature theand viscometer arerange 0–138,000.00 kPa the temperature toallowable 190 USA) witha ahigh-pressure measurable 0.02–10,000.00 cP and and an accuracy of ±1.0%PVT, FS.up The AZ,of USA), andofhigh-temperature viscometer (VISCOlab CVI, Boston, MA, and temperature ofused the are the 0–138,000.00 kPaaccuracy and the up to 190 °C,samples during In addition, thepressure gas humidifier was to avoid oftemperature water the core USA) with a measurable range of viscometer 0.02–10,000.00 cPvaporization and an of ±1.0%inFS. The allowable respectively. In addition, humidifierare was0–138,000.00 used to avoidkPa the vaporization of water in the pressure and temperature of the thegas viscometer and the temperature up core to 190 °C, the displacement experiments. samples during the displacement experiments. respectively. In addition, the gas humidifier was used to avoid the vaporization of water in the core samples during the displacement experiments.
Figure 1. Schematic diagram of experimental apparatuses: (1) Syringe pump; (2), (4), (11) Valve; (3) Syntheticdiagram brine cylinder; (12), (15), (16) Pressure transducer;(1) (6) Nitrogen (7) (2), High (4), (11) Valve; Figure 1. Schematic of (5), experimental apparatuses: Syringecylinder; pump; pressure reducing valve; (8) Gas mass flow controller; (9), (21) Data acquisition system; (10) Gas (3) Synthetic brine cylinder; (5), (12), (15), (16) Pressure transducer; (6) Nitrogen cylinder; (7) High humidifier; (13) Core holder; (14) Confining pressure pump; (17) Electric heating thermostat; (18) FigureDryer; 1. Schematic diagram ofGas experimental apparatuses:(9), (1) Syringe pump;acquisition (2), (4), (11) Valve; (3) (10) Gas pressure reducing valve; (8) Gas flow (21) Data system; (19) Balance; and (20)mass mass flow controller; meter. Synthetic brine cylinder; (5), (12), (15), (16) Pressure transducer; (6) Nitrogen cylinder; (7) High humidifier; (13) Core holder; (14) Confining pressure pump; (17) Electric heating thermostat; (18) Dryer; pressure valve; (8) was Gas measured mass flow by controller; (9),analytical (21) Databalance acquisition (10)Viet Gas Thereducing produced water using an (MEsystem; 204, Sao (19) Balance; and (20) Gas mass Technologies Co. Ltd., Hoflow Chi Minh City, Vietnam) a maximum capacity 220 g and an humidifier; (13) Core holder; (14)meter. Confining pressure with pump; (17) Electric heatingofthermostat; (18)
accuracy 0.0001 g. The water the core samples can be accordingly calculated with the Dryer; (19)of Balance; and (20) Gassaturation mass flowinmeter. measured weight of the produced water. Moreover, the mentioned gas mass flow meter has been The produced water was animage analytical balance (ME 204, Sao Viet Technologies usedproduced to measure the measured produced gas by flowusing rate.by The of the whole experimental has been The water was measured using an analytical balance system (ME 204, Sao Viet illustrated in Figure 2. Ltd., Ho Chi Minh City, Vietnam) with a maximum capacity of 220 g and an accuracy of 0.0001 g.
Co. Technologies Co. Ltd., Ho Chi Minh City, Vietnam) with a maximum capacity of 220 g and an The wateraccuracy saturation in the core samples caninbe calculated withcalculated the measured of 0.0001 g. The water saturation theaccordingly core samples can be accordingly with the weight of measured weight of the produced water. Moreover, the mentioned gas mass hastobeen the produced water. Moreover, the mentioned gas mass flow meter has flow beenmeter used measure the used to measure the produced gas flow rate. The image of the whole experimental system has been produced gas flow rate. The image of the whole experimental system has been illustrated in Figure 2. illustrated in Figure 2.
Figure 2. Image of displacement experiment system.
Figure 2. Image of displacement experiment system.
Figure 2. Image of displacement experiment system.
2.4. Experimental Procedure In this section, steady-state and unsteady-state flow experiments have been used to measure the Klinkenberg permeability and investigate the slippage effect on gas–water relative permeability,
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2.4. Experimental Procedure
respectively.InIn an attempt to achieve steady-state flow, the pressure at inlet and outlet and the flow this section, steady-state and unsteady-state flow experiments have been used to measure the rate at the outlet arepermeability necessary to stable forthe 30 slippage min for effect each measurement point.permeability, Injection pressure Klinkenberg andbeinvestigate on gas–water relative and confining pressure simultaneously increasing average pressure overand thethecore respectively. In anwere attempt to achieve steady-state flow, when the pressure at inlet and outlet flowsamples rate at the outlet are necessary be stable for 30 minKlinkenberg for each measurement point. Injection pressure under was required to be raised, or vice to versa. Regarding permeability measurements and confining pressure were simultaneously increasing average over the core HT condition, the core samples were required to be heated when for five hourspressure at a given temperature (i.e., samples was required to be raised, or vice versa. Regarding Klinkenberg permeability ◦ 90 C) in the temperature control system. The heating process is to guarantee that the temperature measurements under HT◦condition, the core samples were required to be heated for five hours at a of core samples can reach 9090 C the measurements. In addition, the gas viscosity used in the given temperature (i.e., °C)prior in thetotemperature control system. The heating process is to guarantee permeability calculation was required tocan be reach correspondingly with respect to the the experimental that the temperature of core samples 90 °C prior to adjusted the measurements. In addition, gas viscosity used in gas the permeability calculation was as required to betemperature correspondingly adjustedinwith temperature since the viscosity would increase elevated as shown Figure 3a. respect the experimental temperature since the gas viscosity would increase as elevated In contrast, the to viscosity of the synthetic brine rarely changes with temperature as shown in Figure 3b. temperature as shown in Figure 3a. In contrast, the viscosity of the synthetic brine rarely changes As for HTHCP Klinkenberg permeability measurements, the confining pressure would be gently with temperature as shown in Figure 3b. As for HTHCP Klinkenberg permeability measurements, increased to eliminate the effect of stress sensitivity. the confining pressure would be gently increased to eliminate the effect of stress sensitivity.
(a)
(b)
Figure 3. Sensitivity of (a) nitrogen and (b) synthetic brine viscosity to pressure and temperature.
Figure 3. Sensitivity of (a) nitrogen and (b) synthetic brine viscosity to pressure and temperature. By taking two-phase displacement experiments as an example, the detailed procedures of the flow experiments have been introduced asas follows: By steady-state taking two-phase displacement experiments an example, the detailed procedures of the steady-state flow experiments introduced as follows: (1) Prepare the synthetic have brine been and nitrogen as mentioned in Section 2.2, and inject the synthetic
(1) (2) (3)
(4)
(5)
(6)
(7)
brine into the cylinder;
Prepare the synthetic brine and as mentioned in Section the30.0 synthetic (2) Vacuum the core sample for 12nitrogen h and saturate it by injecting synthetic2.2, brineand intoinject it under brine into cylinder; MPathe for 12 h; (3) Place sample the core holder and increase experimental Vacuum the the coresaturated sample core for 12 h andinto saturate it by injecting synthetic brine intotemperature it under 30.0 MPa through the temperature control system to programmed temperature. Confining pressure has for 12 h; also been loaded in this step since the confining pressure is crucial for accurately estimating Place the saturated core sample into the core holder and increase experimental temperature reliable permeability of tight sandstone under reservoir conditions [20]; through the temperature control system topressure programmed Confining pressure has (4) Inject the synthetic brine until the injected is stable temperature. and record the pressure at the inlet also been since the as confining crucial for accurately estimating and loaded outlet of in thethis corestep sample as well the rate ofpressure producedisliquid for determining absolute of the sample; reliablepermeability permeability ofcore tight sandstone under reservoir conditions [20]; (5) With an outlet pressure of 0.10 MPa, the nitrogen is injected until no more brine is produced Inject the synthetic brine until the injected pressure is stable and record the pressure at the inlet and the gas flow rate and injection pressure are recorded for calculating the effective gas and outlet of the core sample as well as the rate of produced liquid for determining absolute permeability at an irreducible water saturation ( ). As such, the endpoints of the relative permeability of thecurve, corei.e., sample; permeability [ , ( )] and [ , ( ) = 0], can be obtained; (6) an Byoutlet considering the of injection pressure being lessisthan 4.50 until MPa (see GB/T brine 28912-2012), the With pressure 0.10 MPa, the nitrogen injected no more is produced and brine gas with specific rate ratio are injected until the flow stable. Two the gassynthetic flow rate andand injection pressure are recorded for calculating the iseffective gascriteria permeability are employed to determine whether reached orofnot: injection pressure at curve, at an irreducible water saturation (Swi ).the Asstable such,flow the is endpoints the(a) relative permeability the inlet keeps steady and (b) the gas flow rate at the outlet is constant; i.e., [Swi ,krg (Swi )] and [Swi ,krw (Swi ) = 0], can be obtained; By considering the injection pressure being less than 4.50 MPa (see GB/T 28912-2012), the synthetic brine and gas with specific rate ratio are injected until the flow is stable. Two criteria are employed to determine whether the stable flow is reached or not: (a) injection pressure at the inlet keeps steady and (b) the gas flow rate at the outlet is constant; After the stable state is reached, the injection pressure, gas, and water flow rate are recorded for determining the corresponding effective gas phase and water phase permeability, respectively.
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(8)
(9)
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Note that the gas and water viscosity, used in Equations (2) and (3), is dynamically varied according to experimental pressure and temperature as shown in Figure 3a,b. The water saturation at current circumstance can be determined by weighting the produced water volume. In this case, two points, i.e., [Swj ,krw (Swj )] and [Swj ,krg (Swj )] (j = 1), on the relative permeability curves can be obtained; More points, i.e., [Swj ,kr (Swj )] (j = 2, 3, . . . , n), on the gas–water relative permeability curves can be determined by dynamically changing the ratio of water flow rate to gas flow rate and repeating Steps 6–7; and, Terminate the experiment when gas relative permeability is less than 0.005 (see GB/T 28912-2012).
Since the steady-state displacement experiments are time-consuming, unsteady-state displacement experiments have been conducted to analyze the sensitivities of the gas–water relative permeability to k g , confining pressure, and temperature. The pre-processing of the unsteady-state displacement experiments is identical to that of the steady-state displacement experiments. In another word, Steps 1–3 in the procedure of the steady-state displacement experiments can be inherited and also used in the unsteady-state displacement experiments. After Steps 1–4, the procedure of the unsteady-state experiments follows: (a) inject the nitrogen using the same injection pressure of the synthetic brine as that in the previous step; (b) record the pressure at the inlet and outlet of the core samples, the volume of produced liquid with equal time intervals during the displacement processes; and (c) terminate the displacement experiments until there is no more brine produced from the core samples. 2.5. Determination of Relative Permeability With the measured data of steady-state and unsteady-state displacement experiments, the gas–water relative permeability can be numerically calculated. The water saturation is obtained by Swj =
m j − mo × 100% m − mo
(1)
where Swj represents the water saturation under a certain condition, j = 1, 2, . . . , n,; m is the total mass of wet core sample saturated with any synthetic brine, g; mo is the mass of dry core sample, g; m j indicates the mass of wet core sample under a certain condition, g. Note that the dynamic mass of the wet core samples can be obtained by subtracting the total mass of the wet core samples by the mass of produced synthetic brine. The gas saturation can be readily achieved with known water saturation since the sum of them is equal to 1.0. The effective permeability of gas and water phase can be respectively calculated by Li and Horne (2001) [22] 2p a q g µ g L k eg = × 103 (2) A( p21 − p22 ) k ew =
qw µw L × 103 A ( p1 − p2 )
(3)
where k eg is the effective gas phase permeability measured with nitrogen, mD; k ew is the effective water phase permeability, mD; p a indicates atmosphere pressure, 0.1 MPa; q g represents the gas flow rate, mL/s; µ g is the gas viscosity under experimental conditions, cP; L is the core sample length, cm; A is the cross area of core sample, cm2 ; p1 indicates the pressure at inlet of core sample, 0.1 MPa; p2 indicates the pressure at outlet of core sample, 0.1 MPa; qw represents the water flow rate, mL/s; µw is the water viscosity under experimental conditions, cP. Considering dynamic fluid pressure and temperature during the experiments, the dependence of fluid viscosity on pressure and temperature have been experimentally measured and shown in Figure 3. It is worthwhile to point out that the synthetic brine viscosity is independent of pressure while the nitrogen viscosity can be slightly increased with increasing pressure at a given temperature. With respect to the pressure and temperature conditions in
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the experiments, the corresponding fluid viscosity can be determined, which can be used to calculate the effective gas and water phase permeability using Equations (2) and (3). Subsequently, the gas–water 2018, 11, x FOR PEER REVIEW 7 of 14 relative Energies permeability can be achieved as out that the synthetic brine viscosity is independent of pressure while the nitrogen viscosity can be k eg (Swj ) slightly increased with increasing pressure krg (Satwja)given = temperature. With respect to the pressure and k abs temperature conditions in the experiments, the corresponding fluid viscosity can be determined, which can be used to calculate the effective gas and water phase permeability using Equations (2) k ew (Swj ) be achieved as and (3). Subsequently, the gas–water relative permeability can
krw (Swj ) = (
)=
( k abs )
(4)
(5)
(4)
where krg is the gas relative permeability; krw represents the water relative permeability; k abs indicates the absolute permeability measured by using synthetic brine, mD. As demonstrated in Equations (4) ( ) ) =to normalize the gas–water relative permeability (5) and (5), the absolute permeability has been(used for the core samples. is the gas relative permeability; represents the water relative permeability; where indicates the absolute permeability measured by using synthetic brine, mD. As demonstrated in 3. Results and Discussions Equations (4) and (5), the absolute permeability has been used to normalize the gas–water relative permeability for the core samples.
3.1. New Correlations: k s vs. k g
3. Results and Considering theDiscussions slippage effect associated with gas flow in tight formation, Klinkenberg (1941) proposed an equation to correlate the gas-measured permeability (k g ) under different mean pressure 3.1. New Correlations: vs. over the core samples (pm ) [11]. Considering the slippage effect associated with gasbflow in tight formation, Klinkenberg (1941) = k s (1 + permeability ) (6) proposed an equation to correlate thek ggas-measured ( ) under different mean pm pressure over the core samples (
) [11].
where k s is the Klinkenberg permeability of core samples, mD; b is the slippage factor, its unit depends (1 + ) at two ends of the core sample. (6) on the unit of the mean pressure; pm is mean of=pressures Equation (6) has demonstrated that the Klinkenberg permeability is linearly related to the reciprocal of mean is the Klinkenberg permeability of core samples, mD; is the slippage factor, its unit where pressure, i.e., 1/p . unit of the mean pressure; depends onmthe is mean of pressures at two ends of the core sample. Considering temperature high effective stress of tight reservoirs, Klinkenberg Equation (6) high has demonstrated thatand the Klinkenberg permeability is linearly related to thethe reciprocal of mean pressure, i.e., 1/ . permeabilities of core samples #1–8 have been respectively examined under three experimental Consideringambient high temperature and HT highcondition effective stress Klinkenberg conditions including condition, (pcp of= tight 2.35reservoirs, MPa, T the = 90 ), and HTHCP permeabilities of core samples #1–8 have been respectively examined under three experimental condition (pcp = 30.00 MPa, T = 90 ). The Klinkenberg permeabilities of core samples #9–12 in conditions including ambient condition, HT condition ( = 2.35 MPa, = 90 ℃), and HTHCP Table 1 have been( measured under=ambient conditions. The experimental results of the coreinsamples condition = 30.00 MPa, 90 ℃). The Klinkenberg permeabilities of core samples #9–12 #1–6 under the ambient, HT, and HTHCP conditions have been used as examples and illustrated by Table 1 have been measured under ambient conditions. The experimental results of the core samples circles, triangles, squaresHT, in Figure 4, respectively. According (6), illustrated the interception of #1–6 under and the ambient, and HTHCP conditions have been usedto asEquation examples and by circles, triangles, and squares in Figure 4, respectively. According to Equation (6), the interception of the regression curves on the vertical coordinates is corresponding to the Klinkenberg permeability. theof regression curves on the vertical coordinates is corresponding to the Klinkenberg permeability. The slope the regression curves moderately represents the magnitude of the slippage factor since The slope of the regression curves moderately represents the magnitude of the slippage factor since the slope is equal to k s × b and the k s is a constant. the slope is equal to
×
and the
is a constant.
(a)
(b)
Figure 4. Cont.
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(c)
(d)
(e)
(f)
Figure 4. Experimental results of Klinkenberg correction for core samples: (a) #1; (b) #2; (c) #3; (d) #4;
Figure 4. results of Klinkenberg correction for core samples: (a) #1; (b) #2; (c) #3; (d) #4; (e)Experimental #5; (f) #6. (e) #5; (f) #6. It can be found that good linear relationships are associated with the experimental data shown in Figure 4. Moreover, the comparison of experimental results under the ambient and HT conditions It can be found that good linear relationships are associated with the experimental data shown in has revealed that the Klinkenberg permeability is independent of the experimental temperature Figure 4. Moreover, the comparison of experimental underIntheaddition, ambientthe andgas-measured HT conditions has while the slippage factor increases as elevated results temperature. revealedpermeability that the Klinkenberg permeability is independent ofescalated. the experimental temperature while the increases when the experimental temperature is This is attributed to the fact the increases raised temperature contributes to theInincreased free path of the gas, which is slippagethat factor as elevated temperature. addition,mean the gas-measured permeability increases positively proportional to the gas-measured permeability. Furthermore, effect temperature on when the experimental temperature is escalated. This is attributed to thethe fact thatofthe raised temperature the gas-measured permeability distinctly the absolute permeability decreases. to the contributes to the increased meanis free pathstrengthened of the gas,aswhich is positively proportional Such findings are corresponding to experimental results reported in the literature [21,22]. In gas-measured permeability. Furthermore, the effect of temperature on the gas-measured permeability comparison to the temperature, the increasing confining pressure affects both the Klinkenberg is distinctly strengthened as slippage the absolute permeability Such findings are corresponding to permeability and the factor. In addition,decreases. the Klinkenberg permeability has been experimental resultsreduced reported in an theincreased literatureconfining [21,22]. pressure In comparison to the compression temperature, the increasing dramatically with since intense collapses partial pores and throats inside core samples,permeability which also contributes the inhibited slippage confining pressure affects both thethe Klinkenberg and the to slippage factor. In addition, factor. the Klinkenberg permeability has been dramatically reduced with an increased confining pressure Besides the previous findings associated with the Klinkenberg permeability and the slippage since intense compression collapses partial pores and throats inside the core samples, which also factor, the Klinkenberg permeability has been found to be linearly related to the under the contributes to the inhibited slippage factor. aforementioned three experimental conditions, which shares similarity with the Klinkenberg ∗ Besides the previous findings with[23]. the Klinkenberg permeability and the slippage factor, permeability measured in tightassociated oil formations Figure 5 shows the relationships between the Klinkenberg permeability has been found to beand linearly related to the high k g under the aforementioned and Klinkenberg permeability of samples 1–12 samples 1–8 under confining pressure condition, respectively. Because Klinkenberg permeability independentpermeability of the temperature, three experimental conditions, whichthe shares similarity with the is Klinkenberg measured in ∗ ∗ the squares in Figure 5 represent the relationship between the Klinkenberg permeability and tight oil formations [23]. Figure 5 shows the relationships between k g and Klinkenberg permeability under both of ambient and high-temperature conditions. A similarly linear relationship under high of samples 1–12 and samples 1–8 under high confining pressure condition, respectively. Because the confining pressure conditions has been depicted as circles in Figure 5. Since ∗ represents the
Klinkenberg permeability is independent of the temperature, the squares in Figure 5 represent the relationship between the Klinkenberg permeability and k∗g under both of ambient and high-temperature conditions. A similarly linear relationship under high confining pressure conditions has been depicted as circles in Figure 5. Since k∗g represents the k g under a specific condition, the linear correlation under specific conditions can be generally represented by a mathematical formulation k s = 0.5915k g , R2 = 0.9423
(7)
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under a specific condition, the linear correlation under specific conditions can be generally represented by a mathematical formulation Energies 2018, 11, 467
= 0.5915
,
= 0.9423
9 of(7) 14
Equation (7) can moderately represent the permeability characteristic of the formation surrounding the aforementioned tight gas wells. Such a relationship provides a convenient way to Equation (7) can moderately represent the permeability characteristic of the formation surrounding estimate the Klinkenberg permeability . As for the high confining pressure condition, the aforementioned tight gas wells. with Suchgiven a relationship provides a convenient way to estimate the Klinkenberg permeability given 5k gcan . Asbe forexpressed the high confining pressure condition, the linearity the linearity among the circleswith in Figure by among the circles in Figure 5 can be expressed by
= 0.0914
,
= 0.9018
(8)
2
ks = 0.0914k R = 0.9018 that the changes of (8) The relatively small slope in Equation (8) hasg ,demonstrated resulted from the increased confining pressure are much larger than that induced by the escalated The relatively small slope in Equation (8) has demonstrated that the changes of k g resulted from temperature. From a microscopic point of view, the increased confining pressure is prone to collapse the increased confining pressure are much larger than that induced by the escalated temperature. partially interconnected pores that are the main contributions of the permeability. It is interesting From a microscopic point of view, the increased confining pressure is prone to collapse partially that the linear relationship between Klinkenberg permeability and hasItbeen well maintained interconnected pores that are thethe main contributions of the permeability. is interesting that the for various experimental conditions. In other words, such and a linear might be an linear relationship between the Klinkenberg permeability k g hascorrelation been well maintained forintrinsic various property of the conditions. core samples, although thea linear coefficients of this depend onof the experimental In other words, such correlation mightcorrelation be an intrinsic property the experimental conditions. core samples, although the coefficients of this correlation depend on the experimental conditions. The significances of Equations (7) (7) andand (8)(8) areare that thethe Klinkenberg The significances of Equations that Klinkenbergpermeability permeabilityof ofcore core samples samples can be determined, provided thatthat thethe coefficients ofofthem canreadily be readily determined, provided coefficients themcan canbe bepreliminarily preliminarily obtained obtained by usingusing representative corecore samples in the same area. AsAs forfor the representative samples in the same area. thegas–water gas–waterflow, flow,the the convenient convenient and accurate description of how to calibrate effective gas phasepermeability permeabilityinvolving involving the the slippage accurate description of how to calibrate thethe effective gas phase is still unavailable. Now,Equations Equations (7) and applied to calibrate the effective gas phase effecteffect is still unavailable. Now, and(8) (8)can canbebe applied to calibrate the effective gas permeability measured in the in experiments determining gas–water gas–water relative permeability, although they phase permeability measured the experiments determining relative permeability, are achieved single-phase flow experiments. although they areinachieved in single-phase flow experiments.
conditions. FigureFigure 5. Correlations between the Klinkenberg permeability andand ∗k∗under 5. Correlations between the Klinkenberg permeability under different different conditions. g
3.2. Calibration of Gas Relative Permeability 3.2. Calibration of Gas Relative Permeability The The gas–water relative permeability hashas been measured the gas–water relative permeability been measuredfor forthe thecore coresample sample #4 #4 by by using using the steady-state displacement experiments under the ambient condition. More specifically, a confining steady-state displacement experiments under the ambient condition. More specifically, a pressure of 2.35 MPaMPa waswas applied during thethe displacement process pressure of 2.35 applied during displacement processwhile whilethe theoutlet outletwas was constrained constrained by a by pressure of 0.10 MPa. With thethe measured pressure and production a pressure of 0.10 MPa. With measured pressure and productiondata, data,the thegas–water gas–water relative permeability of core sample been obtainedthrough throughEquations Equations (2)–(5) (2)–(5) and 6. permeability of core sample #4 #4 hashas been obtained anddepicted depictedininFigure Figure minimize the the effect points measured during experiments, regression curves curves are usedare to 6. ToTominimize effectofofsingular singular points measured during experiments, regression the variation of relative permeabilities. usedpresent to present the variation of relative permeabilities.
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Figure 6. 6. Gas–water Gas–water relative relative permeability permeability with with and and without without Klinkenberg Klinkenberg correction. Figure correction.
The uncalibrated gas relative permeability permeability at the irreducible water saturation is larger than 1.0 as shown in Figure 6. This means that the effective effective gas gas phase phase permeability permeability at at Swi == 33.3% 33.3% is larger than the Equation (7)(7) was utilized to the absolute absolute permeability permeabilitydue duetotothe theslippage slippageeffect. effect.Subsequently, Subsequently, Equation was utilized correct thethe effective gas to correct effective gasphase phasepermeability permeabilityand anddetermine determinethe the Klinkenberg Klinkenberg permeability permeability under different water saturations. The The corrected corrected gas gas relative relative permeability permeability has has also also been been plotted plotted in in Figure Figure 6. 6. It is worthy to note that the gas gas relative relative permeability at the irreducible water saturation reduces to be less than 1.0 after the correction. This phenomenon demonstrates demonstrates that the gas relative permeability may be overestimated if the slippage effect is not considered during the process of determining the relative permeability. permeability. 3.3. Gas–Water Relative Permeability Sensitivity Analysis
With the assistance of the unsteady-state unsteady-state displacement experiments and the newly proposed correlations, the sensitivity analyses areare conducted to investigate the impacts of k g , confining pressure, correlations, the sensitivity analyses conducted to investigate the impacts of , confining and temperature on the gas–water relative permeability. As shown in Table a total eight pressure, and temperature on the gas–water relative permeability. As shown in 2, Table 2, aof total of unsteady-state displacement experiments have been conducted on five core samples in Group 2. eight unsteady-state displacement experiments have been conducted on five core samples in Group The core samples #13–15 areare used forfor evaluating thethe effect of kofg while corecore samples #16 #16 andand #17 #17 are 2. The core samples #13–15 used evaluating effect while samples employed to identify the influence of confining pressure and temperature on the gas–water relative are employed to identify the influence of confining pressure and temperature on the gas–water permeability, respectively. relative permeability, respectively. Table 2. Experimental conditions Table 2. Experimental conditions for for the the sensitivity sensitivity analysis analysis experiments experiments Core Core No. No. 13 13 14 14 15 15
16 16
17 17
∗
∗ ◦C ,kmD Confining Pressure, Confining Pressure,MPa MPa Temperature, Temperature, °C g , mD 0.1418 30 30 9090 0.1418 0.2986 30 90 0.2986 30 90 0.7312 30 90 0.7312 30 90 30 90 2.8730 30 50 9090 2.8730 50 90 30 25 30 25 0.2280 30 50 30 50 0.2280 30 30 9090
Pressure PressureDrop, Drop, MPa MPa 88 8 8 8 8 4 44 4 4 4 4 4 44
3.3.1. 3.3.1. Effect Effect of of k g In an attempt attemptto toinvestigate investigatethe theeffect effect , the confining pressure temperature In an ofof k g , the confining pressure andand temperature havehave beenbeen kept kept constant. The gas–water permeability with k different has been experimentally constant. The gas–water relative relative permeability with different g has been experimentally obtained and obtained and shown Figure 7. Note the effective gas phasecorrespondingly permeability has been shown in Figure 7. Noteinthat the effective gasthat phase permeability has been corrected correspondingly corrected by usingwith the respect developed correlations with respect to the be experimental by using the developed correlations to the experimental conditions. It can found that conditions. It can be found that both the irreducible water saturation and the critical gas saturation (i.e., the minimum gas saturation when the gas starts to flow) decrease due to the increased
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both the irreducible water saturation and the critical gas saturation (i.e., the minimum gas saturation Energies 2018, 11, x FOR PEER REVIEW 11 of 14 when the gas starts to flow) decrease due to the increased permeabilities (i.e., k g ), resulting in an permeabilities (i.e.,water ), resulting an expanded mobile waterpermeability range. In addition, thegas water relative expanded mobile range. Inin addition, the water relative at critical saturation Energies 2018, 11, x FOR PEER REVIEW 11 of 14 permeability at critical gas permeability saturation is relatively low is relatively large for high cores andlarge smallfor forhigh low permeability permeabilitycores cores.and Thesmall thin for section permeability cores. The thin section analyses results of the core samples #13 and #15 have been analyses results of the core samples #13 and #15 have been collected and shown in Figure 8. As can permeabilities (i.e., ), resulting in an expanded mobile water range. In addition, the water relative be collected and shown inconnectivity Figure 8. Asofcan be seen, the pores–throats connectivity of core seen, the pores–throats the core sample #15 much better than thatand of the the core permeability at critical gas saturation is relatively large for is high permeability cores small forsample low #15permeability isSuch much better than that of section the coreanalyses sample #13. Such apermeability difference makes its contribution to not #13. a difference makes its contribution to not only thethe but#13 also the flow capacity of cores. The thin results of core samples and #15 have been only the permeability also the capacity ofthe water and gas saturated in the structure. water and gas the pore structure. collected andsaturated shownbut ininFigure 8. flow As can be seen, pores–throats connectivity ofpore the core sample #15The is much better than that of thecan core sample #13. Such a difference makes contribution toextent. not gas permeability by by water saturation to its atodifferent extent. The gas relative relative permeability canbe beinterfered interfered water saturation a different only the permeability butpermeability also the flow capacity ofirreducible water and gas saturated in the structure. dependence of gas phase onon thethe irreducible water saturation has been addressed The dependence of gas phase permeability water saturation haspore been addressed by gas relative permeability interfered by water different extent. Thefor Wall et etThe al. (1982) (1982) [20]. They Wall al. [20]. indicatedcan thatbewater saturation maysaturation be one of to thea most critical factors dependence of gas phase permeability on the irreducible water saturation has been addressed recovery.Rushing Rushingetetal.al.(2003) (2003)measured measured the effect connate water saturation (5–40%) onbygas the gas recovery. the effect of of connate water saturation (5–40%) on the et al. (1982) [20]. They indicated that water saturation may be one intends of the most critical for as gasWall relative permeability and demonstrated that gas relative permeability intends to factors decrease relative permeability and demonstrated that gas relative permeability to decrease as connate gas saturation recovery. Rushing al. (2003) measured the of effect ofonconnate water saturation on the as connate water saturation The effect mobile water the gas relative(5–40%) permeability water increasesetincreases [18]. The [18]. effect of mobile water the gason relative permeability as shown in gas relative permeability and demonstrated that gas relative permeability intends to decrease shown 7inillustrates Figure 7 illustrates the same phenomenon well. The gas relative permeability of the as core Figure the same phenomenon as well. Theasgas relative permeability of the core sample #13 connate water saturation increases [18]. The effect of mobile water on the gas relative permeability as sample is that higher than that of the#15 core #15 at the same water saturation, although of is higher#13 than of the core sample at sample the same water saturation, although k g of core sample #13 shown in Figure 7 illustrates the same phenomenon as well. The gas relative permeability of the core is lower than#13 thatisoflower the core sample #15. because#15. more mobile water is involved in water the core core sample than that of theThis coreissample This is because more mobile is sample #13 is higher than that of the core sample #15 at the same water saturation, although of sample #15 hassample lower irreducible saturation. It demonstrates that theItreduced gas relative involved inwhich the core #15 which water has lower irreducible water saturation. demonstrates that core sample #13 is lower than that of the core sample #15. This is because more mobile water is permeability presence of mobile water is much larger thatwater caused the irreducible the reduced by gasthe relative permeability by the presence of than mobile is by much larger thanwater that involved in the core sample #15 which has lower irreducible water saturation. It demonstrates that caused by the irreducible water even if their values are identical. even if their values are identical. the reduced gas relative permeability by the presence of mobile water is much larger than that caused by the irreducible water even if their values are identical.
Figure permeability. Figure 7. 7. Effect Effect of of permeability permeability on on gas–water gas–water relative relative permeability. Figure 7. Effect of permeability on gas–water relative permeability.
(a) (a)
(b) (b)
Figure Pore structureillustrated illustratedby bythin thin section section analysis #13 and (b)(b) core Figure 8. 8. Pore structure analysis results resultsofof(a) (a)core coresample sample #13 and core Figure 8. Pore structure illustrated by thin section analysis results of (a) core sample #13 and (b) core sample #15. sample #15. #15. sample
3.3.2. Effect ConfiningPressure Pressure 3.3.2. Effect ofof Confining Figure 9 illustratesthe thedifferent different responses responses of of gas toto confining Figure 9 illustrates gas and and water waterrelative relativepermeability permeability confining pressure changes. Although a high confining pressure may lead to dramatic permeability reduction,
pressure changes. Although a high confining pressure may lead to dramatic permeability reduction,
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Figure 9 illustrates the different responses of gas and water relative permeability to confining pressure changes. a high confining pressure may lead to dramatic reduction, the gas and water Although relative permeability are impacted in different manners. permeability Firstly, the irreducible the gassaturation and water relative under permeability are impacted in different manners. the irreducible water increases the condition of a larger confining pressure.Firstly, This means partially water saturation increases under the condition of a larger confining pressure. This means partially interconnected pores may be closed due to lower throat size and higher capillary pressure, interconnected pores may be closed due to lower throat size and higher capillary pressure, generating generating more irreducible water locked in the core samples. Secondly, the critical gas saturation moreincreases irreducible water locked in confining the core samples. Secondly, the critical gas saturation increases also with the increased pressure. Such a change is attributed to the also reduced pore with the increased confining pressure. Such a change is attributed to the reduced pore and throat size and throat size as well. However, the sensitivity of water relative permeability to the confining as well. However, the sensitivity of water relative permeability to the confining pressure is stronger pressure is stronger than that of the gas relative permeability, resulting in that the reduction of than thatwater of thephase gas relative permeability, that theof reduction water phase effective permeability is largerresulting than the in reduction effective of gaseffective phase permeability, permeability is larger than the reduction of effective gas phase permeability, which is consistent with which is consistent with the changes of gas and water relative permeability resulted from increasing the confining changes ofpressure. gas and water relative permeability resulted from increasingincreases the confining pressure. the As shown in Figure 9, the gas relative permeability and the water As shown in Figure 9, the gas relative permeability increases and the water relative permeability relative permeability decreases while the confining pressure is increased from 30.0 MPa to 50.0 MPa. decreases confining pressure is of increased from MPa water to 50.0 saturation MPa. This isand in line with gas the This is inwhile line the with the observations changes in 30.0 connate critical observations of changes in connate water saturation and critical gas saturation. saturation.
Figure Figure 9. 9. Effect Effectof ofconfining confining pressure pressure on on gas–water gas–water relative relative permeability. permeability.
3.3.3. 3.3.3. Effect Effect of of Temperature Temperature According temperature According to to Figure Figure 3, 3, the the viscosity viscosity of of the the synthetic synthetic brine brine is is more more sensitive sensitive to to temperature compared to the nitrogen. The viscosity of the synthetic brine may reduce by two-thirds compared to the nitrogen. The viscosity of the synthetic brine may reduce by two-thirds while while the the viscosity of nitrogen increases by approximately 20%, provided that the temperature is raised up viscosity of nitrogen increases by approximately 20%, provided that the temperature is raised up from from 25 °C to 90 °C. The gas–water relative permeability at various temperatures has been ◦ ◦ 25 C to 90 C. The gas–water relative permeability at various temperatures has been experimentally experimentally measured depicted Figure 10. effective gas phase permeability measured and depicted in and Figure 10. Theineffective gasThe phase permeability decreases and thedecreases effective and the effective water phase permeability increases due to distinguishable viscosity changes of the water phase permeability increases due to distinguishable viscosity changes of the nitrogen and the nitrogen and the synthetic water with respect to temperature. In addition, the irreducible water synthetic water with respect to temperature. In addition, the irreducible water saturation reduces saturation reduces when the experimental temperature is raised Thiswater generates mobile when the experimental temperature is raised up. This generates moreup. mobile in the more core samples, water in the core samples, resulting in further reduction of the effective permeability as resulting in further reduction of the effective permeability as aforementioned. aforementioned. It has been addressed that the Klinkenberg permeability is independent of experimental temperatures if there is only connate water in the core samples [21,22]. The effective gas phase permeability might be influenced by the experimental temperature during the two-phase displacement experiments, although the temperature-induced differences between gas relative permeabilities are smaller than that between water relative permeabilities. This demonstrates that it is imperative and essential to consider the effect of temperature on the gas–water relative permeability measurements.
Figure 10. Effect of temperature on gas–water relative permeability.
experimentally measured and depicted in Figure 10. The effective gas phase permeability decreases and the effective water phase permeability increases due to distinguishable viscosity changes of the nitrogen and the synthetic water with respect to temperature. In addition, the irreducible water saturation reduces when the experimental temperature is raised up. This generates more mobile Energies 2018, 11, 467 13 of 14 water in the core samples, resulting in further reduction of the effective permeability as aforementioned.
Figure 10. Effect Effect of temperature on gas–water relative permeability. permeability.
4. Conclusions The gas–water relative permeability has been calibrated by considering the slippage effect under various conditions. New linear correlations have been proposed to determine the Klinkenberg permeability by using gas-measured permeability at ambient, high-temperature, and high-temperature– high-confining-pressure conditions. The effects of experimental conditions can be illustrated by the coefficients of the linear correlation. Note that one correlation has been used for describing the linear relationship for both the ambient and high-temperature conditions since the Klinkenberg permeability is independent of the temperature. In addition, it has been demonstrated that the gas relative permeability may be overestimated if the slippage effect is not handled during the process of determining the effective gas phase permeability. With the assistance of the newly proposed correlations, the sensitivities of gas–water relative permeability have been experimentally investigated by considering various gas permeabilities, experimental temperatures, and confining pressures. It has been found that various gas-measured permeabilities lead to different irreducible water saturation, critical gas saturation, and mobile water saturation, which dramatically affect the effective gas and water phase permeability. It is also demonstrated that the reduced gas relative permeability by the presence of mobile water is much larger than that caused by the irreducible water even if their values are identical. Moreover, the sensitivity of water relative permeability to the confining pressure is stronger than that of the gas relative permeability. The gas relative permeability decreases and the water relative permeability increases, provided that the temperature is increased. This is attributed to the synergistic effect of the fluid viscosity and the slippage factor, implying that it is imperative and essential to consider the effect of temperature on the gas–water relative permeability measurements. Acknowledgments: The authors gratefully acknowledge the financial support from National Natural Science Foundation of China (Grant No. 51404282). Author Contributions: Guangfeng Liu and Yang Lu conceived and designed the experiments; Bo Feng, Yu Xia, and Qimeng Zhao cooperatively performed the experiments; Zhaoqi Fan and Siying Li analyzed the data and wrote the paper. Conflicts of Interest: The authors declare no conflict of interest.
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