Zelin Xu ,Gang Li ,*,Xiaosen Li ,**,Hao Chen ,Qiunan Lv ,Changwen Xiao

a Guangzhou Institute of Energy Conversion,Chinese Academy of Sciences,Guangzhou,510640,PR China

b University of Chinese Academy of Sciences,Beijing,100049,PR China

c State Key Laboratory of Hydraulic Engineering Simulation and Safety,Tianjin University,Tianjin,300072,PR China

ABSTRACT The permeability is one of the intrinsic parameters that determines the fluid flow in the porous media.The permeability in hydrate-bearing sediments affects the gas recovery and production of hydrate reservoirs significantly.The irregular permeability characteristics are challenging for fine-grained hydratebearing sediments.In this study,a series of experiments was conducted using an one-dimensional pressure vessel to investigate the hydrate formation characteristics and the permeability in hydratebearing fine quartz sands(volume weighted mean diameter was 36.695 μm).Hydrate saturations(0-26% in volume)were controlled and calculated precisely based on the amount of injected water and gas,the system pressure and temperature.The results indicated that the hydrate nucleation induction period was completed during gas injection,and the average time of hydrate formation was within 500 min.The permeability of methane hydrate-bearing fine quartz sands was investigated by steady gas volume flow.For hydrate saturation lower than 13.94%,the hydrate mostly formed in grain-coating,the permeability reduction exponent calculated by Parallel Capillary,Kozeny Grain Coats and Simple Cubic Filling models were 2.00,2.10 and 1.74 respectively,and Simple Cubic Filling model was in accordance with the experimental data best.However,for hydrate saturation ranged from 13.94% to 25.91%,the permeability increased due to the flocculation structure formation of fine quartz sands and hydrate,which caused the increase of effective porosity.A new relationship among hydrate saturations,effective porosity,the ratio of permeability in the presence and the absence of hydrate was developed.This study developed the mathematical models for predicting the permeability with hydrate saturation in fine quartz sands,which could be valuable for understanding the characteristics of hydrate-bearing finegrained sediments.

Keywords:Methane hydrate Fine quartz sands Hydrate saturation Permeability Mathematical model

1.Introduction

Natural gas hydrates(NGHs)are important potential energy sources.NGHs are formed by natural gas and water molecules at low temperature and high pressure[1].NGHs have attracted global attentions because of the characteristics of high energy density,wide distribution and shallow burial characteristics[2].It was estimated the amount carbon contents in NGHs is twice the sum carbon contents of coal,oil and natural gas.NGHs in nature distribute mainly in the deep sea sediments and permafrost area[3,4].The permeability of the hydrate-bearing sediments is the key factor for gas production and the exploitation efficiency[5,6].The factors affected the permeability of the sediments included the types of the porous media,grain size,porosity of the sediments,hydrate saturation(SH),effective stress and hydrate growth mode[7-9].The grain size of porous media was one of the most important factors[10].

Many researchers studied the water permeability of the hydrate-bearing coarse-grained hydrate sediments.Li et al.[11]measured the water effective permeability in hydrate-bearing quartz sands of 200-300,300-450,450-600 μm.They found that forSHlower than 10%,the formation of hydrate was in the pore center,and forSHhigher than 10%,hydrate packed the sands grain uniformly.Shen et al.[12]used glass beads primarily in the 80-155 μm grain size,and measured the water effective permeability of the hydrate-bearing sediments.They found that the bestfitting model was based on pore filling type.Delli et al.[13]measured the water permeability of the CO2hydrate-bearing Ottawa 20/30 sands of 600-850 μm.They developed a hybrid model,which suggested the hydrate formation morphology from cementing to that of pore filling with increasingSH.Chen et al.[14]investigated water permeability of differentSHin glass beads with diameter ranging 105-125 μm,they found the water permeability firstly decreased sharply forSHwas lower than 3%,then decreased slowly forSHwas higher than 3%.

The permeability reduction models of hydrate-bearing sediments have been studied.Masuda et al.[15]introduced the most widely used model betweenSHand the permeability,the model put forward an adjustable exponent parameterN,which was calculated from the experimental data.Kleinberg et al.[16]summarized a basic equation,the ratio of absolute permeability and the intrinsic permeability was formulated as a function ofSHin quartz sands,the Archies saturation exponentn,and the ratio of the internal surface area of the pore space without and with hydrate.Li et al.[11]assumed hydrate formed in the pore center based on the cylindrical model forSHlower than 10%,and hydrate packed uniformly on the simple cubic sphere and distributed in homogeneous forSHhigher than 10%,then they developed a new relationship between the ratio of the permeability in the presence and absence of hydrate and the hydrate saturation.Shen et al.[12]assumed hydrate grew in the pattern of wedge tetrahedral filling in glass beads,and extended Kleinberg model to an expressionwith more accurate prediction.For the above researches focused on the permeability experiments and numerical simulation in coarse-grained sediments,the relationship between hydrate saturation and permeability is thorough investigation.In free water environment,hydrate formed in the pore center,the permeability decreased sharply then gently.In the numerical simulation,the reduction exponent in different models shows the growth habit of hydrate.However,the majority of the sediments were fine-grained hydrate-bearing sediments.

In recent years,the fine-grained hydrate-bearing sediments of low permeability had been widely concerned[17].The analysis of hydrate sediments samples showed that the hydrate reservoir was mostly consisted by sands,clay and silt,and the particle size of the porous media ranged from 8 to 75 μm[18].In the Ulleung Basin Gas Hydrate expedition in Korea,the sediment samples grain-size distribution is analyzed by a laser-scattering particle analyzer,and the mean grain size lies between 4 and 25 μm,with sparsely scattered coarse-grained intervals[19,20].The first exploitation trial in South China Sea in 2017 showed the sediments were mainly medium-fine silt and coarse silt whose particle size varied from 8 to 32 μm and 32-63 μm[21],respectively.The second gas hydrate trial production were carried out in the Shenhu sea area of the South China Sea,the particle size of the silt sediments ranged from 7.89 to 25.76 μm,which resulted in low permeability[22].The fine particles of the sediments lead to the instability pore structure,sands production and the low permeability of the sediments,which cause much difficulties in the hydrate exploitation.

The laboratory flow experiments in fine-grained sediments were conducted in recent years.Li et al.[8]found that the method of steady-state water injection was no longer recommended for silt with the volume weighted mean diameter of 30 μm,considering the water was hard to flow through the highSH(SH＞20%)sediments.Wu et al.[23]studied the relationship between the methane hydrate saturation and the permeability in montmorillonite sediments by steady gas volume flow,the results showed a decreased permeability with the decomposition of methane hydrate because of the swelling of the clay.Liu et al.[24]found that the permeability in kaolin clay was very different,it showed that the permeability of the kaolin clay firstly decreases and then increases with the increase of hydrate saturation,which caused by the effect of blockage of hydrate particles and aggregation of clay particles.Lu et al.[25]studied the water permeability of hydrate silty-clay reservoirs,and they found the dynamic permeability of the gas hydrate reservoirs presented an increase-decrease-increase trend with a decrease in the gas hydrate saturation.Johnson et al.[26]measured the gaswater relative permeability by steady nitrogen gas flow of fine sands(31-125 μm)from the Mount Elbert site on the Alaska North Slope,where they found that hydrate saturations from 1.5%to 36%could significantly reduce the permeability of porous media.Choi et al.[27]developed a new experimental system and measured the permeability of the brine-saturated hydrate-bearing(SH= 42.0%)specimen(The mixture of sand and kaolinite clay)by unsteadystate gas flow,they met the challenge for maintaining the hydrate stable,and they found the higher residual brine saturations in hydrate-bearing specimen compared with hydrate-free specimen.However,the reported research on the permeability of fine-grained hydrate-bearing sediments is still inadequate.Furthermore,the corresponding mathematical models for the irregularity of the permeability with the hydrate saturation were less.

In this work,we focused on the relationship between permeability and methane hydrate in methane hydrate-bearing fine quartz sands.We used one-dimensional pressure vessel for the hydrate formation and the permeability measurement.The characteristics of hydrate formation in fine quartz sands was investigated.In the permeability measurement,the steady gas volume flow is considered to be an effective method for permeability measurement in hydrate-bearing fine quartz sands.The permeability of unconsolidated fine quartz sands with different methane hydrate saturation was measured by steady gas volume flow.The relationship between the permeability and hydrate saturation was divided into two expressions.In lower hydrate saturations,the characteristics of the permeability with hydrate saturation was discussed in different mathematical models.In higher hydrate saturations,the relationship between methane hydrate saturation and permeability was developed.

2.Materials and methods

2.1.Apparatus

Fig.1 shows the schematic design of the experimental apparatus.The apparatus and experimental setup were designed for hydrate formation and permeability measurement within the fine quartz sands.

The pressure vessel was the key place for hydrate formation and permeability measurement.The pressure vessel was made of 316 stainless steel and could withstand the maximum pressure up to 25 MPa.The inner wall of the pressure vessel was a rough surface to avoid the flow channel between the sediments and the wall in the permeability measurement period.The effective volume of the pressure vessel was 108.48 ml measured by injecting water into the empty vessel.The temperature sensor(-15 to 90°C,±0.1°C)labeled as T was installed in the middle of the pressure vessel.Two pressure sensors(NAT8251.34.2517,0-25 MPa,0.25%)labeled asP1andP2were set at the inlet and outlet respectively of the pressure vessel to record the pressure.The temperature sensor and pressure sensors were from Trafag Company in Switzerland.The differential pressure transducer(STD730-EIAC4AS,0-400 kPa,0.25%)was used to record the differential pressure between the inlet and outlet of the pressure vessel during the permeability measurement,it was from Honeywell Company in America.The gas tank with the volume of 251.50 ml provided methane gas for hydrate formation and permeability measurement,it was made of 316 stainless steel and had a maximum pressure of 34 MPa.The gas flow controller(F221 M,30-1500 ml/min,±0.5%)in front of the pressure vessel could control a steady gas flow through the hydrate sediments in the permeability measurement,it was from Bronkhorst High-Tech B.V in Netherlands.The pressure sensor P3and temperature sensor T3were connected to the gas tank,which had the same type withP1and T,respectively.The coiled pipe with a length of 5 m was used to precool the inject water.The pressure vessel,the gas tank and the coiled pipe were all placed in the water bath(-5 to 20°C,±0.1°C)showed in the dashed box in Fig.1.

Besides the apparatus mentioned above,there were metering pump,electronic balance,back pressure regulator and the data acquisition system.The metering pump(HPLC3000A,0-50 ml/min,±0.01 ml/min,30 MPa)was used to inject water with a constant rate into the pressure vessel,it was from Beijing innovation Tongheng Co.,Ltd..The electronic balance(BS2202S,0-2200 g,±0.01 g)was used to record the weight of water pumped into the vessel,it was from Sartorius Company in Germany.The back pressure regulator(0-25 MPa,±0.02 MPa)was installed at the outlet of the pressure vessel to maintain a constant pressure of the vessel in the permeability measurement.

The pressure sensors,the temperature sensor,the differential pressure transducer,and the gas flow controller were connected to the data acquisition system.The reading time was 5 min and 10 s during hydrate formation and permeability measurement,respectively.

2.2.Materials

Methane gas with the purity of 99.9% was provided by Guangzhou Shengying Chemical Co.,Ltd.,China.The deionized water used for sands cleaning and hydrate formation with a resistivity of 18.25 mΩ/cm was prepared by the ultrapure water equipment provided by Nanjing Ultrapure Water Technology Co.,Ltd.China.

Fig.2 shows the particle size distribution of the fine quartz sands measured by laser particle size analyzer,Mastersizer 2000E,produced by Malvern Instrument Ltd.England.The volume weighted mean diameter was 36.695 μm,and the median diameter D(0.5)was 32.977 μm.The density of the fine quartz sands was 2.589 g/m3,measured by Density Analyzer ULTRAPYC 1200e.The fine quartz sands were used in our study.

2.3.Procedures

The fine quartz sands were cleaned and dried,and then packed into the pressure vessel.Then the pressure vessel with the unconsolidated fine quartz sands was placed in the water bath.The methane gas flow rate of 30 ml/min was injected into the pressure vessel for 10 min to remove the air in it.Appropriate amount of water was driven into the pressure vessel and stood for two days for uniformly distribution in the pressure vessel.Then the outlet valve was closed and the pressure vessel was pressurized by methane gas for hydrate formation at the working temperature.The temperature of the water bath was set at approximately 4°C.Methane hydrate began to form in fine quartz sands in the pressure vessel,and pressures of inlet and outlet began to decrease until there was no change.The gas,water and hydrate saturations could be calculated for the whole hydrate formation period.

There was the permeability measurement after hydrate formation.The back pressure regulator was set to the endpoint pressure after hydrate formation of each run.Steady methane gas flowed through the pressure vessel to measure the permeability of the hydrate-bearing fine quartz sands.

The porosity of the sediments can be calculated as equation(1):

Wherems= 151.38 g is the weight of fine quartz sands filled into pressure vessel.ρs=2.59 g/cm3is the density of fine quartz sands.V= 108.48 ml is the total volume of the pressure vessel.The porosity of the fine quartz sands was calculated to be φ0= 45.98%.

The mass of the injected water is given by the mass of water formed methane hydrateand by the mass of water remained in the pressure vessel(SWVpρW)as equation(2):

WheremWis the mass(g)of injected water.SHandSWare the aimed methane hydrate saturation and the water saturation,respectively.NH= 5.75 is the hydration number of the hydrate.MW= 18 g/mol andMH= 119.5 g/mol are the molar masses of water,respectively.Vp=49.88 ml is the pore volume of the pressure vessel.ρW=1.00 g/cm3and ρH= 0.94 g/cm3are the densities of water and hydrate,respectively.

Take Run 4 for example.The methane hydrate saturationSH= 10%,and the water saturationSW= 7%.The mass of injection water was calculated to be 7.55g.

The derivation of Darcy's law for gas was illustrated in Fig.1.The pressure and the volume of a given mass of confined gas are inversely proportional at 4°C according to Boyle's Law[28]as equation(3):

Where,P1andP2are pressures(MPa)at the inlet and outlet of the pressure vessel,respectively.P is the average pressure(MPa)ofP1andP2.P0=0.1 MPa is atmosphere pressure.Q1,Q2,QandQ0are the gas volume flow(ml/min)under the condition ofP1,P2,PandP0,respectively.

According to Darcy's law[29]as equation(4):

Qcould be calculated as equation(5),which was combined equation(3)and equation(4).

Where,KGis the permeability measured by gas(md).ΔPis the differential pressure(MPa)between inlet and outlet pressures.A= 3.14 × 10-4m2is the cross section of the pressure vessel.L=0.3455 m is the length of the pressure vessel.μ is the dynamic viscosity(Pa·s)of methane under the condition ofPand 4°C.

From equation(5),KGcan be obtained as equation(6).

Table 1 shows the dynamic viscosity of specific pressure and temperature[30].The dynamic viscosity μ of methane under the condition ofPand 4°C in this study can be fitted as equation(7).

Table 1 The viscosity of methane gas from reference.

3.Results and discussion

3.1.Hydrate formation in fine quartz sands

With changes of the system pressure and temperature with time,SH,SGandSWcan be calculated as following(8)-(11)equations[31]in the hydrate formation process:

Where,vmis the molar volume(ml/mol)of the methane gas under the conditions of current temperature and pressure.nm,Gis the molar amount(mol)of the remained methane remaining after hydrate formation.nm,G0is the initial molar amount(mol)of methane.mWis the mass(g)of the injected water.

Table 2 provides the initial pressure and results for the formation of methane hydrate.Considering the similarity of procedures and features for all experiments in this study,we chose Run 4(SH= 10.01%)as a typical one to illustrate the pressure profiles during hydrate formation and the permeability measurement.

Table 2 Formation conditions and results:Initial pressures P1 and P2,P1 and P2 after hydrate formation,Methane hydrate saturation(SH),Water saturation(SW),Gas saturation(SG)and synthesis periods(t).

Fig.3 shows the evolution of the inlet pressure,outlet pressure and saturations of each phase during methane hydrate formation in Run 4.The induction period of methane hydrate was not observed because of the induction period was complete while methane flowed into the pressure vessel.The initial pressures of inlet and outlet were 10.24 MPa,and then decreased rapidly because of the formation of methane hydrate.The hydrate formation process lasted for 393 min,and the inlet and outlet pressures finally kept constant of 8.93 MPa.The finalSHreached 10.01%,and the gas and water saturations decreased from 84.30% to 82.44% and 15.70%-7.56%,respectively.The time of the hydrate formation period lasted for about 500 min in average.The rapid formation characteristic was different from previous work[32].A possible explanation was that the fine quartz sands had larger the surface area than coarse ones[33],and the water distributed uniformly and fully contacted with methane.

3.2.The permeability measurement

The permeability measurement of differentSHin fine quartz sands was carried out after the methane hydrate formation.In Run4,the pressure of the vessel was set to 6.70 MPa by the back pressure regulator.Then the steady-state methane flowed through the hydrate-bearing sediments with steady inlet and outlet pressures.

Fig.4 provides the inlet and outlet pressures and differential pressure changes with time for the permeability measurement in Run 4.It was assumed that very limited hydrate dissociated,and theSHmaintained constant during the experiments of permeability measurement.The reasons for the above assumption were as follows:(1)The inlet and outlet pressures(6.70 MPa)were much higher than the equilibrium pressure(3.60 MPa)at the working temperature(4°C).(2)The differential pressure was almost constant at 7.99 kPa,which indicated the methane hydrate didn't decompose during the steady gas flow measurement in the sediments.Once the hydrate decomposition happened,the differential pressure would decrease rapidly.(3)It can be seen that the permeability measurement process just lasted for 400 s.

Table 3 shows the conditions and results of the permeability measurement.There were totally 18 runs included 4 additional runs in Run 0 and Run 3 to study the permeability characteristics in hydrate-bearing fine quartz sands.

Table 3 Experimental results of hydrate saturation(SH)after hydrate formation,inlet(P1)and outlet(P2)pressures,differential pressure,gas volume flow rate(Q0),the dynamic viscosity of methane(μ)and the permeability(KG)at 4 °C during permeability measurement.

In runs 0,the value of permeability wasn't affected by different inlet and outlet pressures and gas volume flow.We also tested randomly in Run 3,which had the same results as in Run 0.According to equation(5),the ratio ofQ0andis a constant.is equal to(P1-P2)(P1+P2),where(P1-P2)is the differential pressure.So when(P1+P2)is a constant,the increase ofQ0will result in the increase of differential pressure(P1-P2).But the smaller gas volume flow may result in low accuracy of the differential pressure transducer,the greater gas volume flow may result in gas breaking in the hydrate-bearing sediments.So the control of the gas volume flow was important and difficult.The gas volume flow was set to 200 ml/min in our study with a flexible adjustment around it.

The permeabilityKGof fine quartz sands changed in two stages:it firstly decreased slightly then increased with the increase ofSHfrom 0 to 13.94% and 13.94% to 25.91%,respectively.

Fig.5 provides the possible explanations for the permeability reduction and growth characteristic.Fig.5(a)is the initial unconsolidated fine quartz sands.The particle size of the fine quartz sands varied from 0.363 μm to 131.862 μm.Fig.5(b)shows the grain-coating hydrate formed in fine quartz sands forSHincreased from 0 to 13.94%,which resulted in the permeability of a slight decrease from 142.05 md to 91.54 md.ForSHfrom 13.94%to 25.91%showed in Fig.5(c),the small fine quartz sands(＜10 μm)and methane hydrate collided to form the flocculation structures[34].Then the good gas flow channel was formed.The results were similar to the results of pervious studies[23,24,35].

3.3.Kr-SH relationship with SH lower than 13.94%

Methane hydrate tend to form on grain surfaces in free gas environment[36].The similar slow permeability reduction characteristics with the increase ofSHbetween our experimental results and hydrate coat grains models should be taken in to consideration.Therefore,we analyzed the regularity of the permeability in lowerhydrate saturation in our study by three mathematical models for hydrate coating grains.

Parallel Capillary model[16]assumed that the porous medium consists of a bundle of straight,parallel cylindrical capillaries,with the methane hydrate uniformly coated the walls of each capillary.The ratio of the permeability in the presence and the absence of hydrateKrcan be calculated as equation(12):

Where,K0=142.05 md is the reference permeability whenSHwas lower than 13.94%.Equation(12)correlates the data withR2= 0.6790.R2is the correlation coefficient.R2is the ratio of Residual Sum of Squares and Total Sum of Squares.

Kozeny Grain Coats model[16]describes the hydrate coating the grain surfaces in the granular media as equation(13):

Where,nis a parameter that determines the decreasing rate of permeability withSH.

Equation(13)correlates the data with the least-squares nonlinear regression coefficients ofR2= 0.8107.The best correlation of theKrvariation bySHwas obtained asn= 2.1055.

Shen et al.[12]described hydrate coat grains in Simple Cubic Filling mode,the Simple Cubic Filling model assumed the minimum repeatable unit was a cube with the grain in it,and hydrate formed by coating grains till the cube inscribed in the sphere hydrate.SHwas described as equation(14),andKrcould be expressed as equation(15):

Equation(15)correlates the data with the least-squares nonlinear regression coefficients ofR2= 0.8979.The best correlation of theKrvariation bySHwas obtained asn= 1.7403.

Fig.6 provides the comparison of the experimental data with the three theory models.The results showed that experimental data agreed with Simple Cubic Filling model best forSHfrom 0 to 13.94%.The Simple Cubic Filling model was more accurate in predicting the relationship betweenKrandSHthan Parallel Capillary and Kozeny Grain Coats models.Methane hydrate formed by grain coats in fine quartz sands occurs in gas free environment rather than hydrate occupied the pore center.However,the values of the permeability in Runs 2,3,4 and 5 were less than that from the theory models.This is because a small fraction of methane hydrate occupied pore centers,causing lower permeability than theoretical value.

As theSHincreased from 0 to 13.94%,the effective porosity(φe)decreased from 45.98% to 39.57%.The effective porosity is defined as the volume fraction occupied by fluid phases(gas and water phases)as equation(16)[37].

3.4.Kr-SH relationship with SH higher than 13.94%

The flocculent structure of fine quartz sands and methane hydrate formed withSHincreased from 13.94% to 25.91%.It was assumed that the effective porosity increased because of the formation of the flocculent structure.The permeability of the sediments significantly increased from 91.54 md to 603.76 md.We tested that methane didn't flush the water out whenSW＜7.56%,in other words,the remained water in fine quartz sands after hydrate formation was not the gas flow obstruction.Therefore the measured permeability was actually the absolute permeability[38].The value of permeability was just affected bySHin our study.

Wu et al.[35]applied the Ives and Pienvichitr model and Tien's model to analyze the relationship betweenSHand φeof Illite hydrate-bearing sediments,the results showed that theSHincreased from 18.89% to 31.16%,φeincreased from 37.7% to 38.9%and 39.5%-40.6%,respectively.Considering the similar lower water absorption characteristics between Illite and fine quartz sands,the mathematical description of theSHand φeof fine quartz sands in our study can be expressed as equation(17):

Where,φeis the effective porosity.a= 0.0036 is the first-order coefficient.b= 0.3952 is a constant.

As long as methane hydrate saturation was between 13.94%and 25.91%,the effective porosity can be accurate calculated through equation(17).

Civan model[39]provided the relationship between porosity and permeability variation as equation(18)in porous media undergoing scale dissolution and precipitation processes,and they correlated experimental data with levels of accuracies sufficient for practical applications.

Where,f∞is a constant value.K0is the reference permeability(md).φris the reference porosity.Kis the permeability(md).φ is the porosity.β is the lumped parameter.

For the hydrate saturationSHchanged from 13.94%to 25.91%,we employed the Civan model in our study.TheSH= 13.94% was the reference saturation start point.Therefore,the constantf∞=1,and the reference porosity φ0=39.57%.Thus,the relationship between effective porosity and the permeability in our study could be simplified to equation(19).

Where,KGis the permeability measured by gas flow(md).K0= 91.54 md is the reference permeability whenSHwas higher than 13.94%.β is an exponent determining the increasing rate of the permeability because effective porosity changed.Equation(19)correlates the data with the least-squares nonlinear regression coefficients ofR2=0.8249.The best correlation of theKrvariation bySHwas obtained as β = 582.12.Then by invoking equation(17)into equation(19),the relationship betweenKGandSHwas obtained.

Fig.7 showed the relationship amongKrandSH,and φe.The value of φehad the linear growth from 39.57% to 39.61% andKGincreased exponentially from 91.54 md to 603.76 md with the increase of methane hydrate saturation from 13.94% to 25.91%.The increase of methane hydrate saturation contributed to form the flocculent structure and the sediments became incompact,the gas was easier to flow through the sediments,and finally resulted in the rise of the permeability.

However,the hydrate core samples consisted of different percentage of sand,silt and clay,which were different from the fine quartz sands in the work.Therefore in the further investigation,it is recommended to study the permeability of the natural hydratebearing sediments by steady gas volume flow.

4.Summary and conclusions

In this study,the mathematical models for predicting the permeability with hydrate saturation in fine quartz sands were established.The permeability of unconsolidated fine quartz sands before and after hydrate formation in the pressure vessel were investigated with methane using Darcy's Law for gas.During permeability measurement,steady gas volume flow and stable differential pressure for minutes were obtained.The conclusions were as follows:

(1)In free gas environment,the hydrate nucleation induction period of hydrate formation was completed during gas injection.The average time of hydrate formation was within 500 min.

(2)For the hydrate saturationSHlower than 13.94%,theKr-SHrelationship was investigated by Parallel Capillary,Kozeny Grain Coats and Simple Cubic Filling models with different reduction exponents.Simple Cubic Filling model fitted best with the experimental data.Most methane hydrate was formed by grain-coating though a small fraction formed by pore-filling.

(3)As hydrate saturation increased from 13.94% to 25.91%,the effective porosity grew linearly from 39.57% to 39.61%,resulting theKrincreased.A new relationship betweenKrandSHwas developed based on the flocculation structure in fine quartz sands.The overallSH-φe-Krrelationship was obtained.AsSHwas within 25.91%,the values of the methane saturations,effective porosity and the permeability were continuous and consistent.

Declaration of competing interest

The authors declare no conflict of interest.

Acknowledgments

This work is supported by National Natural Science Foundation of China(51976228),Key Program of National Natural Science Foundation of China(51736009),Frontier Sciences Key Research Program of the Chinese Academy of Sciences(QYZDBSSWJSC028,QYZDJ-SSW-JSC033)and the Special Project for Marine Economy Development of Guangdong Province(GDME-2020D044)which are gratefully acknowledged.

Nomenclature