Measuring absolute adsorption in porous rocks using oscillatory motions of a spring-mass system

Younki Cho,Ryan Lo,Keerthana Krishnan,Xiaolong Yin,Hossein Kazemi

Petroleum Engineering,Colorado School of Mines,Golden,CO 80401,USA

Keywords:Adsorption Porous media Shale Capillary condensation Oscillation

ABSTRACT We present an oscillation-based method to measure absolute adsorption,or total gas,contained in porous rocks without and with excess adsorption.Experiments conducted with a macroporous Berea sandstone sample in nitrogen where excess adsorption is negligible show that absolute adsorption can be obtained after the added mass of co-accelerated gas outside the sample is subtracted.In experiments conducted in propane with a crushed Niobrara shale sample with micro-and mesopores,absolute adsorption included significant excess adsorption.After subtracting both the added mass outside the sample and the gas that would be in the sample assuming no excess adsorption existed,estimated excess adsorption of propane is in good agreement with that projected based on capillary condensation of propane in the volume of mesopores.

1.Introduction

Quantifying the amount of gas stored in a porous media is important for gas purification and separation processes that involve solid and porous sorbents[1].It is also important for management of geological gas reservoirs,including natural gas fields,underground storage of hydrogen,and permanent geosequestration of carbon dioxide[2–7].Total gas stored in a porous medium is termedabsoluteadsorption [8].Inmacroporousmedia(pore diameter greater than 50 nm),the absolute adsorption is simply the product of the net pore volume and the gas density.Inmesoporous(pore diameter between 2 and 50 nm) andmicroporous(pore diameter less than 2 nm) media,adsorption of gas molecules on pore surface and the subsequent capillary condensation can significantly increase or even dominate the absolute adsorption.Capillary condensation,specifically,is a vapor–liquid phase transition that nucleates in a pore following surface adsorption at a pressure lower than the saturation pressure of bulk gas and has been extensively investigated theoretically and experimentally and recently through molecular simulations [9–12].

In experiments,surface adsorption and capillary condensation are collectively presented as theexcessadsorption that can be quantified using mass,mass ratio,or volume ratio.Excess adsorption expressed in mass is the difference between the absolute adsorption in the porous medium over the mass of gas that would occupy the pores of the medium with bulk density assuming that surface adsorption and capillary condensation were absent [8].Excess mass ratio is the mass of excess adsorption normalized by the mass of the sample.Excess volume ratio is excess mass expressed in volume,usually at the standard condition,normalized by the volume of the sample.

This study’s main subject is porous rock,a class of porous media that usually contains a range of pore sizes.Measuring excess adsorption in shale gas reservoir rocks is particularly important,because these rocks are known to have abundant micro-and mesopores [13].Excess adsorption is not only a tool for extraction of pore size distributions of shale rocks [14] but also a critical part of gas storage[15].Experiments that target pore size distributions are usually conducted at low temperatures and pressures,whereas those that aim to characterize gas storage need to be carried out at reservoir conditions.

Regardless of experimental conditions,most measurements use manometric,volumetric,and gravimetric methods.A manometric/volumetric setup generally consists of a reference cell and a sample cell.The manometric method uses pressures of the cells to estimate transferred gas,whereas the volumetric method uses volumes of the cells controlled by precision volume pumps.Prior to an experiment,the void volume of the sample cell with the porous solid placed inside is determined using a non-sorbing gas.The nonsorbing gas is then removed from the sample cell.With the same porous solid inside,the adsorbing gas is next introduced into the sample cell to begin the adsorption process.In this process,the temperature of the sample cell is kept constant and the amount of the gas transferred into the sample cell is controlled by either the pressure or the volume of the reference cell.The excess adsorption is determined according to Eq.(1)

wheremtrans=Δ（VP）Mw/ZRTandmref=VvoidPMw/ZRT.Here,Pis pressure,Tis temperature,Mwis the molar mass of the adsorbing gas,Ris the universal gas constant,Zis the compressibility factor of the adsorbing gas,Vvoidis the void volume measured by the non-sorbing gas,and Δ（VP）is the volume-pressure change in the reference cell that yields the amount of gas transferred [16].Traditional manometric/volumetric methods have been used in all applications of gas sorption from determination of pore size distribution to characterization of gas storage [17–20].Manometric/volumetric setups can also be extended to perform additional characterizations while measuring sorption.For instance,a gas chromatograph may be connected to the sample cell to determine gas composition when the test is conducted with a gas mixture[21].The specially designed gas adsorption apparatus in Poneet al.[22]can measure sorption in a porous solid with confining stress.Another specially designed volumetric/manometric apparatus by Hol and Spiers [23] contains a displacement sensor and can measure adsorption-induced swelling of porous media.

In gravimetric methods,excess adsorption is obtained by directly weighing the sample and correcting for buoyancy.Among the various gravimetric methods,the magnetic suspension balance method developed by De Weireldet al.[24]to measure adsorption and condensation at high pressures and high temperatures is particularly notable[25,26].In their method,the sample is placed in a chamber filled with gas and held at the condition of interest.The weight change to the sample is measured by an analytical balance located outside the chamber,connected to the sample by a magnetic coupling.The magnetic suspension balance is capable of measuring weight changes up to ±2 μg.A direct comparison between the volumetric method and the magnetic suspension balance method is presented in [16].Other than the magnetic suspension method,the mass change occurred to an immersed sample placed on a balance can also be measured using piezoelectric weight sensors [27].There are other gravimetric devices in the literature.However,they do not measure weight change of an immersed sample on a balance but use weight difference to derive transferred mass,i.e.,mtransin Eq.(1).For instance,Dayet al.[28,29] built a gravimetric device capable of measuring mass change with an accuracy of±1 mg.Their device consists of a sample cell and a reference cell,with each cell placed on its own balance.The mass of the reference cell (no sample but with gas) provides the density of the gas,hence removes the need of an equation of state.The gravimetric device of Barsottiet al.[30] is similar with a higher accuracy (±0.01 mg) however without a reference cell for gas density.

In this study,we present an oscillation-based method to determine the absolute adsorption (total gas) in porous rocks at highpressure conditions.Frequencies of vibrations are mass dependent and have been used to determine mass changes under highpressure and/or temperature conditions.The quartz crystal microbalance(QCM),for instance,uses vibration of a quartz crystal with a frequency in the megahertz range[31,32]and can detect nglevel mass change to a 20 μg sample.Bonner and Cheng [31] first used QCM for measuring solubility of a gas in a polymer under high pressure at 125 °C.Schaefet al.[33] employed QCM to measure carbon dioxide adsorptions using 10 MHz quartz crystals on 7–30 μg of montmorillonites clays at 50 °C and pressure up to about 13 MPa.In addition to QCM,Briscoe and Mahgerefteh [34] also developed a vibrating-reed method to determine gas-polymer interactions at high pressures using a beam with a vibration frequency of about 200 Hz[34,35].The oscillation method used in this study is different from previous vibration-based methods in that it uses a spring-mass system enclosed in a pressure vessel and operates in O(10) Hz range.This method was first presented in Larsonet al.[36] however has not been applied to any porous media.In this study,we applied this method to determine the absolute adsorption first in Berea sandstone,a macroporous rock in which excess adsorption is negligible and the absolute adsorption is simply the product of the pore volume and the gas density,and then in Niobrara shale,a typical shale reservoir rock with micro-and mesopores in which excess adsorption is not negligible.This paper presents the procedure needed to characterize the total and excess adsorptions contained in these porous rocks at various pressures and temperatures and the results.

2.Experimental Setup and Procedure

In this section,we present the experimental setup and procedure.The experimental setup is similar to that in [36].The experimental procedure however includes additional steps to account for the effects of temperature and sample porosity.

2.1.Experimental setup

The experimental setup is presented in Fig.1.A pressure vessel that is 7.62 cm (inner diameter) × 16.51 cm (length) with a pressure rating of 28 MPa (4000 psi) holds the sample and the gas.Sample is placed in a weight holder,suspended by a spring (EI 007A 01 S,Lee Spring)attached to the top cap of the pressure vessel.A cylindrical magnetic rod is attached to the bottom of the weight holder and is inserted into a solenoid (527-1022-ND,Digi-Key).This solenoid is fixed to the base of the pressure vessel and is used to initiate oscillations and measure frequency.A compression seal fitting (TG-24 T(CU)-A2-G,Conax Technologies) allows the solenoid’s wires to exit the pressure vessel.When the solenoid is connected to a 12 V DC power supply,it pulls the sample and the holder downviathe magnetic rod.After that,the solenoid is disconnected from the DC power supply and switched to an oscilloscope(Model 2530B,BK Precision).The return of the sample,the holder,and the magnetic rod to their equilibrium positions triggers an oscillation.The oscillatory motion of the magnetic rod in the solenoid generates a voltage signal,the frequency of which is read from the oscilloscope.

The pressure vessel has one inlet and one outlet.The inlet is connected to a transfer vessel(CP2-GM,Welker,0–34.5 MPa)used to provide gas.The outlet has a valve to relieve pressure and is connected to a vacuum pump(D75,Precision),used to evacuate gas in the pressure vessel and the tubings.Pressure in the pressure vessel was monitored using a digital pressure gauge (DPI 104,GE Druck,0–34.5 MPa).A heating tape (TBSAT051-020,Briskheat,0–232.2°C)was used to supply heat to the pressure vessel.The temperature of the pressure vessel was monitored by a multimeter with a thermocouple (HHM290 TrueRMS SuperMeter,Newport).

2.2.Materials

In this study,nitrogen and propane were selected.Propane is a hydrocarbon component found in many natural gas reservoirs.The critical point of propane is 96.7 °C,4.248 MPa and therefore it is a condensable gas at the conditions of this study.Nitrogen,in contrast,has a critical point of -146.9 °C and 3.390 MPa.It is hence supercritical at the experimental conditions and cannot be condensed.

Two porous rocks were selected.Berea sandstone (Cleveland Quarries,Vermilion,Ohio) is dominated by macropores [37,38],with a porosity of 19.70% and a permeability of 175 md (1 md=9.8692 × 10-16m2).The surface area of Berea sandstone ranges from 0.57 to 1.4 m2?g-1[39–42].Niobrara shale (CEMEX Quarry,Lyons,Colorado)was selected as our mesoporous sample.Niobrara shale in U.S.Rocky Mountain region is a major tight petroleum resource play.The Niobrara shale is a self-sourced reservoir consisting of chalk and marl[43,44].The porosity of the Niobrara shale is about 7.9% while the permeability ranges from 4 × 10-4to 1 × 10-2md [43].The presence of micro-and mesopores in Niobrara shale leads to 4.3–14.4 m2?g-1of surface area [44].Berea sandstone core plug was prepared in the size of 3.810 cm diameter × 3.175 cm length.Fresh sample of Niobrara shale was cleaned by toluene and methanol using a Soxhlet extractor.Then,the sample was crushed and sieved to 20/40 mesh size (425–850 μm) to accelerate gas–solid interaction.Both the core plug and the crushed sample were kept in an oven before and after the experiments to avoid absorption of moisture from the air.

2.3.Principle of measurements

For a solid mass suspended by a Hookian spring and submerged in a stationary fluid,assuming that the drag force is proportional to the velocity of the solid by a constant coefficientc,the equation of motion for the solid is

wheremis the mass sensed by the spring,kis spring constant,andxis the distance of the solid away from the equilibrium position.The solution to the above equation,assuming that att=0,x=xmax,and dx/dt=0,is

Fig.1.Schematic of experimental setup (not drawn to scale).

The above equation indicates that the velocity of the solid has a frequency of and the amplitude of the velocity should decrease as an exponential function oft.The voltage induced in the solenoid is directly proportional to this velocity.c/2mcan thus be estimated from the decay of the amplitude of the voltage signal.Oncec/2mis determined,mis

Althoughmappears on the right-hand side of Eq.(6),it is not treated as an unknown there becausec/2mis directly measurable.When（c/2m）2?k/m,the solution formis simplified to

In our measurements,amplitudes of voltage signals indeed decreased as exponential functions oftand the values ofcranged between 0.00312 and 0.02466 N?s?m-1depending on the pressure.With typicalkandmin the experiments,（c/2m）2?k/mis valid.Thus,we adopted Eq.(7) in all of our calculations.

Mass sensed by the spring,as determined by Eq.(7),consists of the following

In this equation,m0is the original mass of the solid sample,mais the mass of co-accelerated gas outside the solid sample (added mass),andmbis the mass of co-accelerated gas inside the solid sample,the absolute adsorption.mbcan be further divided intompthat simply equals the product of the pore volume of the sample and the density of the gas and Δmthe excess adsorption.In the next section,we present the procedure to determine the above components in the sensed massm.

2.4.Procedure to determine k,m0, ma,mb,and mp

Experiments began with determination of the spring constantkusing a known massmc,following equation

These experiments were carried out under the ambient pressure like in[36].The difference between this study and[36]is that we foundka function of temperature.Experiments to determinekwere conducted at three temperatures (21.1 °C,46.1 °C,65.6 °C).Heating of the pressure vessel took 5–6 hours and was continuously monitored to assure stabilization of the temperatures.

The original massm0was measured using a precision balance(EL 303,Mettler Toledo,± 0.001 g) at the ambient condition.In addition to the mass of the sample,m0also includes the mass of the magnet rod and that of the sample holder,the sum of which is 19.100 g.The mass of the spring,which is 0.018 g,is not included inm0.

The experimental procedure to determinemais the same as that in [36].Specifically,maas the mass of co-accelerated gasoutsidethe sample should be directly proportional to gas density,and,according to previous vibration studies [35,36],to the crosssection area of the sample perpendicular to the direction of oscillation.In this study,Berea sandstone sample is cylindrical and a cylindrical holder was made to contain the crushed Niobrara sample.Hence,non-porous cylinders that have the same cross-sections as the Berea and the contained crushed Niobrara samples were used to determinema.For these non-porous cylinders,mbis zero.Measured deviations inmfromm0are therefore entirely due toma.mawas measured at a number of pressures and three different temperatures (21.1 °C,46.1 °C,65.6 °C) using propane and nitrogen.Prior to these measurements,a vacuum pump was connected to the pressure vessel to degas the sample,the flow lines,and the vessel for 30 minutes.Then,the vacuum pump was disconnected from the pressure vessel and gas was let in using the transfer vessel to start the measurements.For best results,the system was allowed to stabilize for 15–30 minutes between pressure steps.Becausemais proportional to gas density,from the experiments we seek to establish

where α is termed the added-mass coefficient.A linear regression betweenk/4π2f2m0andMwP/ZTm0provided α [36].α depends on the shape of the sample and the gas used.

For porous solids,co-accelerated gas is not only present outside of the solid sample but also inside.This gas in the name of absolute adsorptionmbis divided intompand excess adsorption Δm

where

Vpis the pore volume of the porous solid.Separation ofmbin Eq.(11) is only necessary for the extraction of the excess adsorption Δm.If one is only interested in the absolute adsorptionmb,there is no need to breakmbintompand Δm.mbwas calculated using measuredm0and the just established added mass coefficient

The Berea sandstone sample was tested in nitrogen at 21.1 °C up to 5.5 MPa.As most pores in Berea sandstone are macropores,Δmis negligible.Hence,mbmeasured from the oscillation frequency is practicallymp:

mpemerged from Eq.(14)was compared to that calculated from Eq.(12)usingVpobtained separately from a core measurement system CMSTM-300 (Core Laboratories) for verification.The compressibility factor of nitrogen used was from NIST (National Institute of Standards and Technology) Webbook [45].

For the crushed Niobrara sample,its pore volume cannot be determined using CMSTM-300 and hence we used nitrogen as a non-sorbing gas to determinempand then the pore volumeVp.First,Eq.(14) was used to determinempin nitrogen.Then,Eq.(12) was used to computeVpof the crushed sample.Vpwas then used,again in Eq.(12) but with propane’sMwandZ,to determinempof the Niobrara sample in propane.In the end,the excess adsorption Δmin the experiments conducted with propane is

Determination ofmpand in turnVpof a porous sample using a non-sorbing gas is a common practice in manometric,volumetic,and gravimetric measurements.Helium is generally the choice.In this study,however,due to the low density of helium,we were not able to determinemaandVpusing helium.Nitrogen was therefore used as a substitute.Nitrogen is supercritical under the experimental conditions and hence is non-condensing.Supercritical adsorption of nitrogen in shale is known to be low.Chareonsuppanimitet al.[46] measured nitrogen adsorption in Albany shale at high pressures.Their data suggest that at 55.05°C and in the pressure range of 1.47–12.4 MPa,the quantity of nitrogen adsorbed is only 0.0012–0.0147 mmol?g-1.For Niobrara rocks,no highpressure nitrogen adsorption data are available.Assuming that the quantity of nitrogen adsorbed in Niobrara is comparable to that in Albany shale,excess nitrogen due to adsorption in our experiments with 70–80 g of Niobrara sample should be O(0.01) g level.This is an order of magnitude smaller than the excess adsorption observed in our experiments (will be presented shortly).Hence,in this study we treated nitrogen as non-sorbing.Note that this treatment will result in overestimatedVpand under-measured Δm.

3.Results and Discussion

In this section,we first present spring constantkat different temperatures.Then,we present frequencies at different pressures and temperatures for the non-porous cylinders of the same shapes and sizes of the samples and their added-mass coefficients.Lastly,we present absolute adsorptions measured in the Berea and crushed Niobrara samples.Excess adsorption Δmin the crushed Niobrara sample was extracted and discussed.

3.1.Spring constant k

Table 1 shows the frequencies and spring constants measured at three different temperatures under the ambient pressure.Note frequencies and spring constants presented are averages of six measurements.Decrease in the spring constant was observed with increasing temperature.The reduction in the spring constant with increasing temperature is due to the reduction in the Young’s modulus of the material.

3.2.Determination of ma and α

To determinema,the mass of co-accelerated gasoutsidesolid samples,non-porous cylinders with known masses were tested.These non-porous cylinders have shapes and sizes identical to those of the porous samples.We performed a total of five experiments,summarized in Table 2.

Table 1 Frequencies and spring constants measured using objects with known masses at different temperatures

Table 2 Summary of experiments conducted to characterize ma

For propane gas,pressure was increased to the saturation pressure of the respective temperature,while experiments with nitrogen gas was tested to 10 MPa.Six frequency readings were taken at each data point.The averaged frequencies are shown in Fig.2 with error bars representing the standard deviations of the six readings.The average of the coefficients of variation offis 0.087%indicating the consistency in the measurements.

Fig.2.Frequencies of non-porous cylinders with known masses as functions of pressure for (a) propane and (b) nitrogen.

Fig.3. k/4π2f2m0 as a function of MwP/ZTm0 for(a)propane and(b)nitrogen.Dotted lines are linear regressions of the data with unit intercept.The slopes of the dotted lines are the added-mass coefficients α.

A linear regression line was drawn betweenk/4π2f2m0andMwP/ZTm0to determine α (Fig.3).For the non-porous cylinder simulating the holder of the crushed Niobrara sample,a single α was found from experiments conducted in propane gas at three different temperatures.When the same cylinder was put in nitrogen,a lower α was obtained.α obtained in nitrogen for the nonporous cylinder simulating the Berea sample was lower than that for the Niobrara sample in nitrogen due to difference in their cross-sections.Here,the error in α,δα,from linear regression is calculated as below following [47]

wherenis the number of points used in the linear regression andrepresents average.Results for α and δαare summarized in Table 3.α will be used in the next section to determinempand Δm.

Table 3 Added-mass coefficients and uncertainties of linear regressions

3.3.Absolute adsorption in Berea and Niobrara samples

At 21.1 °C,nitrogen gas was pressurized up to 5.5 MPa,and oscillation frequencies of the Berea sample were measured as the pressure was increased.This experiment was repeated three times and despite some noise consistent trends were produced.As illustrated in Fig.4,the normalized frequency,defined as the frequency obtained at the pressure of measurement divided by the frequency obtained at the ambient pressure,of the three data sets are consistently below the dashed line plotted assuming that the sample had no gas inside using the added mass coefficient α=8.431 × 10-6mol?K?Pa-1established in the previous section.This difference shows the influence of the absolute adsorption inside the sample on the frequencies of oscillations.From these data,mpof the Berea sample was calculated using Eq.(14).

To verifymp,we plottedmpagainstVpMwP/ZTRin Fig.5.The pore volumeVpof 7.24 cm3used in this comparison was from CMSTM-300.Fig.5 shows thatmpfrom oscillations agree withVpMwP/ZTR.This agreement proves thatmpmeasured is indeed the absolute adsorption of the Berea sample.

Fig.4.Normalized frequencies of the Berea sandstone sample in nitrogen as a function of pressure.The dashed line represents the frequency trend assuming that there is no gas inside,plotted using α=8.431 × 10-6 mol?K?Pa-1.

Following the experiments on the Berea sample,oscillation frequencies of crushed Niobrara sample were measured in nitrogen at 21.1°C up to 5.5 MPa.In these experiments,m0=60.524 g and the mass of the crushed Niobrara sample is 41.419 g.Nitrogen was assumed to be non-sorbing.Similar to Fig.5,Fig.6 shows that normalized frequencies collected on the crushed Niobrara shale are consistently below the line assuming that the sample had no gas inside plotted using α=1.290×10-5mol?K?Pa-1.mpwas then calculated according to Eq.(14)and they are presented in Fig.7.Fig.7 shows that when pore volumeVpwas selected to be 7.859 cm3,measuredmpagreed withVpMwP/ZTR.7.859 cm3was thus accepted as the pore volume of the crushed Niobrara sample.This volume includes both the macropore volume between crushed grains and the micro-and mesopore volume within crushed grains.The net porosity of the crushed sample is 33.88% based on grain density of 2.7 g?cm-3.

Fig.5.Verification that mp determined for the Berea sample is the absolute adsorption VpMwP/ZTR.The diagonal line corresponds to y=x.

Fig.6.Normalized frequencies of the crushed Niobrara sample in nitrogen as a function of pressure.The dashed line represents the frequency trend assuming that there is no gas inside,plotted using α=1.290 × 10-5 mol?K?Pa-1.

Fig.7.Relation between mp in crushed Niobrara sample in nitrogen and VpMwP/ZTR when Vp is 7.859 cm3.The solid diagonal line is y=x.

We now present absolute and excess adsorptions obtained from experiments conducted with the crushed Niobrara sample in propane at 21.1 °C.In these experiments,m0was (98.875 ± 0.026) g and the mass of crushed Niobrara sample was (79.770 ± 0.026) g.Absolute adsorptionmbwas calculated using Eq.(13) with added mass coefficient α=2.266 × 10-5mol?K?Pa-1.In propane experiments,we expect excess adsorption because Niobrara shale has abundant micro-and mesopores and,under the condition of the experiments,propane is condensable.Fig.8 shows thatmbincreased with increasing pressure and indeed became greater thanVpMwP/ZTRwhen the relative pressureP/P0exceeded 0.8,indicating excess adsorption.Here,P0=0.8613 MPa is the saturation vapor pressure of propane at 21.1 °C.

Fig.8.Absolute adsorption in the crushed Niobrara sample at 21.1°C.The solid line represents VpMwP/ZTR plotted using Vp=7.895 cm3. P0=0.8613 MPa is the saturation vapor pressure of propane at 21.1 °C.

We will now use the commonly employed Kelvin equation

to interpret the observed excess adsorption in our experiments in the rangeP/P0≥0.8.In Eq.(17),P/P0,the onset of capillary condensation,is a function of pore radiusr(assuming cylindrical pores),contact angle θ of condensed liquid on the pore surface,interfacial tension σ between liquid and gas,liquid density ρ,and temperatureT.Using data from NIST Webbook [45],σ=7.455 mN?m-1and ρ=0.4984 g?cm-3atT=294.2 K,and assuming that the pore surface is completely wetted by liquid propane(θ=0°),one can obtain that the pore radius for capillary condensation to occur atP/P0=0.8 isr=2.4 nm.Note thatr=25 nm whenP/P0=0.979,the highest relative pressure in our experiments.Hence,the excess adsorption shown in Fig.8 can be interpreted as capillary condensation in themesopores(diameter 2–50 nm).

Published pore size distribution data for Niobrara shale[14,43,44,48] all indicated abundant mesopores.Based on combined small-angle neutron scattering,nitrogen adsorption,and mercury inject data,Zhaoet al.[48] reported that the volume of mesopores in their Niobrara shale sample is approximately 1/3 of the total pore volume.The mass of our crushed Niobrara sample is 79.770 g and the density of the grain is 2.7 g?cm-3.Assuming that the porosity inside crushed grains is 7.9%and 1/3 of the porosity is mesoporosity,the volume of mesopores in our Niobrara sample should be 0.78 cm3.At the saturation vapor pressure of propane at 21.1°C,the density of liquid propane is 0.498 g?cm-3and that of gas propane is 0.019 g?cm-3.When gas propane in 0.78 cm3of mesopores turns into liquid propane,there should be a mass increase of 0.37 g.This estimation compares well with the maximum quantity of excess adsorption observed in our experiment,which is about 0.3 g.The excess adsorption observed in this study is hence very well correlated to capillary condensation of propane in the mesopores.

These results show that both the absolute adsorption in the tested Niobrara sample on the order of 0–0.5 g and the excess adsorption on the order of 0–0.3 g can be detected by this oscillation-based measurement method.Normalized by 79.770 g of rock mass,maximum absolute adsorption observed in our experiment is 0.14 mmol?g-1and maximum excess adsorption is 0.08 mmol?g-1.These data are subjected to rather large noise.An error analysis (see Appendix) shows that the observed noise or fluctuation is consistent with expected error propagation.The error in the measured absolute adsorption primarily came from the error in the frequency.Improving the accuracy of frequency measurements should lead to more accurate results in the future.

4.Conclusions

This paper presents a simple experimental method that uses oscillations to measure absolute adsorption in porous media including excess adsorption in micro-and mesopores.Built upon the results of the previous work[36]that the true mass of an object can be determined from frequencies of oscillations by subtracting the mass of co-accelerated gas outside the object,in this study we established that for porous solids,the mass associated with absolute adsorption inside the solid also contributes to reduce the frequency of oscillation,and this mass can be obtained by subtracting the mass of the porous sample and the added mass of coaccelerated gas outside the porous sample from the total sensed mass.

Experiments conducted with a Berea sandstone sample show that for a macroporous medium,measured absolute adsorption is indeed that calculated using the pore volume of the medium and the density of the gas.For a crushed Niobrara sample with micro-and mesopores,we first experimented nitrogen and calculated the pore volume of the sample assuming that nitrogen is non-sorbing.In propane experiments,measured absolute adsorption clearly exceeded that calculated using the pore volume of the Niobrara sample and the density of propane gas when the relative pressure became greater than 0.8.Interpretation based on the Kelvin equation suggests that observed excess adsorption is capillary condensation in the mesopores of the sample.The maximum amount of excess adsorption observed,which is about 0.3 g,is in good agreement with projected excess adsorption based on published pore size distributions of Niobrara shale and densities of fluids.These results demonstrate the potential of this simple method in measuring gas storage in porous solids.

Nomenclature

ccoefficient of friction,N?mm-1?s

ffrequency of oscillation,Hz

kspring constant,N?mm-1

Mwmolecular weight of gas,g?mol-1

mmass,g

mamass of co-accelerated gas outside the sample,g

mbmass of co-accelerated gas inside the sample or absolute adsorption,g

mcmass used in spring constant determination,g

mpmass of stored gas assuming that the gas is non-sorbing,g

mrefmass of reference obtained using a non-sorbing gas,g

mtransmass transferred into the sample cell in manometric/volumetric methods,g

m0original mass of the sample,g

Δmexcess adsorption,g

nnumber of points used in linear regression,dimensionless

Ppressure,MPa

P0saturation vapor pressure,MPa

Runiversal gas constant,8.314 J?K-1?mol-1

R2coefficient of determination for linear regression,dimensionless

rradius of a cylindrical pore,nm

Ttemperature,K or °C

ttime,s

Vppore volume measured separately,cm3

Vvoidvoid (pore) volume measured by a non-sorbing gas,cm3

Δ（VP） volume-pressure change in the reference cell in manometric/volumetric methods

xdistance away from the equilibrium position,m

xmaxinitial amplitude of oscillation,m

Zcompressibility factor of gas,dimensionless

α added-mass coefficient,mol?K?Pa-1

δferror in frequency,Hz

δkerror ink,N?mm-1

δmerror in the mass sensed by the spring,g

δPerror in pressure,MPa

δTerror in temperature,K

δmaerror inma,g

δZerror inZ,dimensionless

θ contact angle,rad

ρ density of liquid,g?cm-3

σ interfacial tension,mN?m-1

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors acknowledge support from the Colorado School of Mines Undergraduate Research Fellowship,the Unconventional Natural Gas and Oil Institute (UNGI) and the Unconventional Reservoir Engineering Project (UREP) consortia at the Colorado School of Mines,and the Chevron International Fellowship.

Appendix

The error in the mass measurements was evaluated by the propagation-of-error method.The contributions of system components to the net error in the measured mass were calculated separately and then summed.In the calculations below,we will use numbers from the Niobrara shale sample experiments conducted in propane to exemplify this procedure.

The error in the spring constant δkis given by

wheremc=98.912 g,f=6.303 Hz,error of the oscilloscope δf=0.01 Hz,and that of the known mass=0.001 g.Based on Eq.(17),the relative error δk/kis 0.32% and is dominated by the term

Then,the error in mass sensed by the spring δmis calculated by

δmfrom Eq.(18)is 0.45 g and the relative error δm/mcis 0.45%.Contributions from δkand δfare nearly equal.However,since δkis dominated by δf,this error is also dominated by δf.

The error in the added mass is calculated using

where α=2.266 × 10-5mol?K?Pa-1,δα=1.6 × 10-7mol?K?Pa-1,Mw=44.096 g?mol-1for propane,P=0.8 MPa,δP=7 kPa (1 psi),Z=0.8476,δZ=0.0003,T=294 K,and δT=1 K.Based on Eq.(19),the error in the added massis 0.04 g.The net relative error/mcis 0.04%.55%of the error is from pressure(δP).Linear regression (δα) contributes about 37%.Error in temperature (δT) contributes about 8%.The error from the gas law (δZ) is negligible.

Lastly,the error in the absolute adsorptionmbis determined using

Fluctuations in the measuredmpwere observed to be 0.1–0.2 g(Fig.8).Hence,in this study we have indeed reached the level of errors permitted by the accuracy of measurements.The primary source of error is the frequency of oscillation and this will be the focus of future improvement.