Research ArticleA Comparative Study on Fracture Characteristics of the RedSandstone under Water and Nitrogen Gas Fracturing

Yanan Gao ,1,2 Feng Gao ,1,2 Zekai Wang,1,2 and Peng Hou1,2

1State Key Laboratory for Geomechanics and Underground Engineering, China University of Mining and Technology,1 Daxue Road, Xuzhou, Jiangsu 221116, China2School of Mechanics and Civil Engineering, China University of Mining and Technology, 1 Daxue Road, Xuzhou,Jiangsu 221116, China

Correspondence should be addressed to Yanan Gao; yngao@cumt.edu.cn

Received 20 March 2018; Revised 12 June 2018; Accepted 24 June 2018; Published 10 July 2018

Academic Editor: Mohsen S. Masoudian

Copyright © 2018 Yanan Gao et al. 'is is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

Because of the disadvantages in fracturing with the water-base fracturing fluids and the development of reservoir reconstructiontechnology, nonaqueous fracturing fluid plays a more and more important role in the worldwide exploitation of unconventionalnatural gas. In this paper, the fracturing experiments of the red sandstone by using water and nitrogen gas are firstly carried out,and the breakdown pressures and failure patterns of the red sandstone specimens under different fracturing fluids are compared.'en, based on the governing equations, the fracturing experiments with water and nitrogen gas are modeled by using a finiteelement method software—COMSOL Multiphysics. 'e conclusions can be obtained as follows: (1) 'e breakdown pressure ofthe nitrogen gas fracturing is 60% that of the water fracturing.'e ultralow viscosity property of nitrogen gas is the reason for thisphenomenon. (2) Compared with the water fracturing, the nitrogen gas fracturing causes greater volumetric strain and a morecomplex fracture pattern in terms of the number, length, and width of the cracks. (3) 'e numerical results are close to theexperimental data. It implies that numerical modeling in this study can be used as a tool for predicting the breakdown pressureand rupture time. (4) After a sensitive study based on the numerical modeling, it can be found that the loading rate will influencethe seepage range which dominates the pore pressure distribution and affects the breakdown pressure for the water fracturing.However, for the nitrogen gas fracturing, the breakdown pressure almost does not change with the loading rate as the nitrogen gascan easily penetrate the specimen from the radial direction.

1. State of the Art

Natural gas is the third largest source of energy in the worldafter coal and oil. Meanwhile, among the total reserve ofnatural gas, the amount of unconventional natural gas is farmore than that of conventional natural gas [1]. 'e devel-opment of unconventional natural gas not only can alleviatethe global energy crisis but also can reduce carbon dioxideemissions and mitigate the global warming [2]. Since theshale gas revolution arose in 2009 [3], more and more at-tentions have been paid to the unconventional gas pro-duction. Because of the richness in unconventional naturalgas resource, such as coalbed methane, tight gas, and shalegas [4, 5], China is also making great efforts to develop the

unconventional gas [6]. However, unconventional naturalgas is difficult to exploit due to the low permeability of theformations, such as shale formation. Soeder indicated thatthe permeability of shale fractures is generally about(0.001∼0.1) × 10−3 μm2 [7]. Javadpour showed that thepermeability of the Shale matrix is 10−9 μm2 orders ofmagnitude [8]. Since the permeability is too low to exploit, itis necessary to develop the reservoir stimulation technolo-gies to improve the permeability of the formations. Now-adays, the most popular and mature method to increase theproductivity in the world is hydraulic fracturing with water-base fracturing fluids. Such technology mentioned above hasplayed a very important role in the shale gas revolution.However, with the development of nature gas engineering,

HindawiAdvances in Civil EngineeringVolume 2018, Article ID 1832431, 15 pageshttps://doi.org/10.1155/2018/1832431

the defects of the fracturing technology with water-basefracturing fluids are exposed gradually, such as (1) waterwasting, especially in the areas of water deficient, for example,the Chinese Ordos area, where the large-scale fracturing withwater-base fluids is difficult to be implemented; (2) water lockeffect and the Jamin effect, which will occur along with thewater injection and result in blocking the gasmigration channeland impairing the productivity [9, 10]; and (3) fracturing fluidbackflow problem, the fluids contain many chemical in-gredients which will cause the unrecoverable damage to thestratum and the underground water pollution [11, 12].

As the drawbacks mentioned above, the fracturing tech-nologies with the water-base fluids have been banned by moreand more organizations or countries [13]. Meanwhile, frac-turing is still indispensable in nowadays gas and oil engi-neering [14]. 'erefore, technological innovation has becomea necessary way for normal unconventional gas production.Nonaqueous fracturing is deemed to be a method that canreplace the fracturing with water-base fluids and to solve thewater wasting, reservoir damage, and the environmentalproblem. Meanwhile, nitrogen gas is regarded as a great kindof fracturing fluid that can be used in nonaqueous fracturing.Compared with the water fracturing, the nitrogen gas frac-turing has the following advantages: (1) the nitrogen reservesrichly in the air, and thus people do not need to pay muchattention to the fracturing fluid resource and the fracturingfluid wasting; (2) nitrogen does not react with montmoril-lonite, and hence the water lock effect and Jamin effect can beavoided; (3) as a gas, nitrogen is easy to discharge, and there isno damage to the reservoir and no gas retention problems; (4)nitrogen can promote desorption of nature gas and increasethe productivity [15]; and (5) nitrogen is a safe fluid as it doesnot react with natural gas.

As early as 1981, Abel used nitrogen gas as a fracturingfluid in the Ohio Shale Formation [16]. 'e results showedthat this technology succeeded in production promotion,even without a propping agent, and it had outperformedother stimulation systems. Wozniak et al. compared thereservoir reform effect of nitrogen gas fracturing, nitrogenfoam fracturing, and mixed nitrogen fracturing and foundout that nitrogen gas fracturing did best in productionpromotion with the minimum cost [17].

To make a further understanding of fracture charac-teristics of the red sandstone under water and nitrogen gasfracturing, in this study, the uniaxial test and Brazilian testare carried out to get the basic parameters of the redsandstone at first. 'en, the comparison study between thewater fracturing experiment and nitrogen gas fracturingexperiment is carried out with the aspects of breakdownpressure, volumetric strain of the specimen, and failurepattern. Afterwards, the numerical method is proposed andvalidated. Finally, the effects of the loading rate in both thewater fracturing experiment and the nitrogen gas fracturingexperiment are also investigated.

2. Experimental Materials and Procedure

2.1. Specimen Preparation. 'e intact sandstone blocks arecollected from the Shandong Province, China. Cutting work

and polishing work of the cylindrical cores are carried out bya diamond cutter with an acute blade and a grinder, re-spectively. Firstly, the specimens are prepared in generalaccordance with ISRM (1978, 1983). 'e height and di-ameter of the test specimens are about 100mm and 50mm.'en, the specimens for the fracturing tests need to beprocessed further. A central borehole of 10mm in diameterand 60mm in length is drilled axially from the midpoint ofthe specimen by using a diamond bit (Figure 1(a)). After-wards, the surfaces and boreholes of the specimens arecleaned by using the clean water which is a transparent,colorless, tasteless, and neutral liquid. After the dimensionsof the prepared specimens are measured, the specimens areplaced in a 100°C oven for 24 hours. 'e dry mass of thespecimens is obtained after they had been cooled down toroom temperature. Finally, the specimens are stored insideheat shrinking PVC jackets.

'rough the SEM test, it can be observed that the grainsof the red sandstone are loosely arranged and pores exist inthe intergrain (Figure 1(b)). Based on the XRD test, it can befound that quartz, feldspar, and mica are the main mineralconstituents of the red sandstone (Figure 1(c)).

2.2. Experiment Equipment. 'e uniaxial compression testsand Brazilian disc tests are carried out by using the TAW-2000 electro-hydraulic servo-controlled rock mechanictesting system with a load capacity of 2000 kN to obtain themechanical parameters of the rocks. 'e gas fracturing testsare performed by using the TAW-2000 electrohydraulicservo-controlled rock mechanic testing system and the high-pressure gas fracturing system (Figure 2). 'e high-pressuregas fracturing system consists of the air compressor, the gastank, the control cabinet, the booster pump, and the mediumgas cylinders.

'e maximum values of the output gas pressure andstandard flow are 80MPa and 63 L/min, respectively. 'edevice, as shown Figure 3, is used to load water pressure onspecimens. It is mainly driven by the electric motor anda spiral metal rod.

2.3. Fracturing Experiment Method. 'e procedure of thefracturing tests is interpreted as follows:

(1) A platen with a double concentric O-ring encirclingthe central injection port is designed (Figure 4(a)).To prevent gas filtration from the specimen bottomor top, a seal gum with high bond strength is usedto seal the specimen bottom and the platen end(Figure 4(b)). Before the test, the time of air-dryingshould not be less than 48 h.

(2) Fracturing tests are carried out at room temperatureand at a uniaxial stress of 4MPa. Any sealing deviceson the sides of the specimen are uninstalled duringthe tests since the fracturing tests are carried out atuniaxial state. Meanwhile, the axial extensometerand lateral extensometer are attached to the speci-men to measure the deformation of the specimenduring water or gas injection

2 Advances in Civil Engineering

(3) For the water fracturing test, the water is injected to thetank of the hydraulic (water) loading device �rstly.�enthe water pressure valve is opened, and the spiral metalrodwill move forward to apply the water pressure to thespecimen until the specimen to be fractured.

(4) For the nitrogen gas fracturing test, �rstly, the aircompressor and cold dryer are opened. �e gas tankis �lled by the compressed air of 1MPa, which is usedas the driving gas.�en, the nitrogen cylinder valve isopened to apply load to the nitrogen gas, which isused as the media gas. Afterwards, the booster pumpsystem is started to apply the gas pressure until thespecimen to be fractured.

It is important to note that the devices for the waterfracturing and gas fracturing are di erent. Meanwhile, theloading mechanisms and control methods of the waterfracturing and gas fracturing are also di erent. �us, it canhardly make the loading conditions the same for those twokinds of fracturing tests, such as loading rate.

2.4. Analysis of Experimental Results

2.4.1. BasicMechanical Parameters. Asmentioned in Section 1,to have an understanding of the basic mechanical prop-erties of the red sandstone in this paper, we �rstly carriedout the uniaxial compression test (Figure 5). �e averagevalues of uniaxial compression strength (UCS), maximumstrain, and Young’s modules are 89.5MPa, 0.005, and 15GPa,respectively.

Afterwards, the �attened Brazilian disk specimens areemployed for testing and evaluating the tensile strength andfracture toughness of the red sandstone. �e thickness anddiameter of the test specimens are about 20mm and 50mm,and the loading angle is about 20° [18, 19].�e failure patternand load-displacement curves are shown in Figures 6 and 7,respectively.

According to the analytical solutions of the tensilestrength and fracture toughness proposed by Wang andXing [20], the tensile strength and fracture toughness can bewritten as follows:

60 m

m

Φ = 10 mm Φ = 50 mm

100

mm

(a) (b)

3000

2500

2000

1500

1000

500

00

MF FFF

F

F

F: feldsparM: mica

Q: quartz

FFF M

M

M MM MF

Q QQ

QQ

Q

Q

10 20 30X-ray diffraction angle (°)

Inte

nsity

(cou

nts)

40 50 60 70

(c)

Figure 1: Preparation and mesoscopic tests of the red sandstone specimen. (a) Specimen design. (b) Scanning electron microscope (SEM)pattern of specimen. (c) Analysis of X-ray di raction.

Advances in Civil Engineering 3

σt � 0.962Pc

πDh, (1)

KIC � 0.7997Pmin��R

√h, (2)

where D, R, and h are the diameter, radius, and thickness ofthe specimen, respectively; Pc is the crucial load, that is, themaximum load during the test and Pmin is the local mini-mum load; and σt and KIC are the tensile strength andfracture toughness of the red sandstone, respectively.

Based on (1) and (2) and the measured parameter of thetests (Table 1), the average values of tensile strength andfracture toughness are 4.5MPa and 0.99MPa·m1/2,respectively.

2.4.2. Fracturing Tests. As shown in Figure 8, the volumetricstrain/water pressure (gas pressure)-time curves of 6 spec-imens during water and nitrogen gas fracturing are given,where W1, W2, and W3 denote the specimens of the waterfracturing, and G1, G2, G3 denote the specimens of nitrogengas fracturing. For the water fracturing, the pressure loadingrates are 0.09552MPa/s, 0.07328MPa/s, and 0.06368MPa/s;the breakdown pressures of the specimens are 7.52MPa,7.38MPa, and 7.75MPa. For the nitrogen gas fracturing, thepressure loading rates are 0.03259MPa/s, 0.02254MPa/s,and 0.01237MPa/s, and the breakdown pressures of thespecimens are 4.0MPa, 4.7MPa, and 4.2MPa.

It can be found that the breakdown pressure of the waterfracturing is higher than that of the nitrogen gas fracturing.�e reason of the phenomenon may be that the breakdownpressure has a strong relationship with the viscosity offracturing �uid. A lower �uid viscosity will result in lowerpressure needed to breakdown the specimen, while a higherfracturing �uid viscosity will require a higher pressure tobreakdown the same specimen [21]. As we know, comparedwith water, nitrogen gas has the property of ultralow vis-cosity. �us, it can breakdown the specimen at lowerpressure. Because there are many existing microfractures orpores inside the specimen, nitrogen gas can more easily

Axial loadingsystem

Loading control system

Confining pressureloading system

Electromagnetic valve

H2O/N2

Aircompressor

Refrigerated drier

Buffer vessel

Air vessel

Recoveryequipment

Flowmeter

Pressure transducer

Pressure pumpTriaxial cell

Specimen with jacket

Figure 2: Testing equipment.

Figure 3: Hydraulic (water) pressure loading system.

4 Advances in Civil Engineering

penetrate the microfractures or pores, which results intransmitting the injection pressure to all the penetratedpoints. �is decreases the strength of the specimen andallows the nitrogen gas to break it at lower pressures [13]. Onthe contrary, the high-viscosity �uid, such as water, tends toremain with the borehole and less likely to be able topenetrate these microfractures. �erefore, a higher pressureis needed to break the specimen by using the high-viscosity�uid. Meanwhile, because the nitrogen gas can transmit theinjection pressure to all the penetrated points, the porepressure of the specimen of nitrogen gas injection is higher

than that of water injection under the same injectionpressure. �e stress intensity factor is positively correlatedwith the pore pressure [22]. �erefore, breaking down thespecimen with nitrogen gas will result in a higher stressintensity factor, and a lower breakdown pressure is needed.

However, for both the water fracturing and the nitrogengas fracturing, the breakdown pressure does not changeobviously with the loading rate, which indicates that threeloading rates of the experiment do not have a signi�cante ect on the breakdown pressure of the water fracturing ornitrogen gas fracturing.

As shown in Tables 2 and 3, volumetric strain under thenitrogen gas fracturing is much larger than the one under thewater fracturing pressure. Note that the volumetric strain ismeasured at the injection pressure of 4MPa as the specimenwill be fractured if the pressure is higher than 4MPa underthe nitrogen gas fracturing.

(a) (b)

Figure 4: Sealing of the specimen. (a) Platen. (b) Specimen bonded with platen.

0.000 0.002 0.004Strain

0.006 0.008

UT-1UT-2UT-3

0

15

30

45

60

75

90

105

Stre

ss (M

Pa)

Figure 5: Stress-strain curves of specimens under uniaxial com-pression. UT-1, UT-2, and UT-3 denote the uniaxial compressiontest of specimens 1, 2, and 3, respectively.

Figure 6: Failure pattern of the �attened Brazilian disk specimen.

Advances in Civil Engineering 5

�e water fracturing and the nitrogen gas fracturingmake di erent failure patterns, as shown in Figure 9, and thefailure pattern of the specimen after the nitrogen gas frac-turing is more complex. �ree main cracks are formed; thecracks propagate through the entire specimen. For eachmain crack, there are several secondary cracks around it.However, for the specimen under the water fracturing, thereare only two symmetrical cracks which do not penetratethrough the longitude (axial) of the whole specimen.Meanwhile, it can be observed that the widths of the cracksformed after nitrogen gas fracturing are also larger thanthose formed after the water fracturing.

3. Numerical Simulation

�e PDE module in COMSOL Multiphysics is used to carryout themodeling of the water fracturing and the nitrogen gasfracturing. By using MATLAB, a seepage stress-couplingalgorithm with the time-expanding seepage boundary isdeveloped. �e calculation routine and �ow chart are shownin Figures 10 and 11.

3.1. Stress Field. According to the principle of e ective stressproposed by Terzaghi and improved by Biot, the relationshipbetween total stress, e ective stress, and pore pressure can bewritten as [23]

σij � σij′ + βpδj, (3)

where σij is the total stress; σij′ denotes the e ectivestress; β is called Biot’s coe§cient; β � 1− (Kb/Ks), Kb isthe bulk modulus of porous media and Ks is the bulkmodulus of solid framework particles; p is the pore pressure;and δj is the identity tensor.

In our fracturing experiments, all specimens are standardcylindrical specimens with a borehole through the mid-point; therefore, the stress �eld can be illustrated as shown inFigure 12.�e equilibrium equation, geometric equations, andphysical equations can be written in polar coordinates.

Without volume force, the equilibrium equation com-bining with principle of e ective stress can be written as

zσr′zr+σr′ − σθ′r

+z(βp)zr

� 0, (4)

where σr′ is the radial e ective stress and σθ′ is the circum-ferential e ective stress. �e physical equations are

σr′ � 2G εr +]

1− 2]εv( ),

σθ′ � 2G εθ +]

1− 2]εv( ),

(5)

where εr is the radial e ective strain, εθ is the circumferentiale ective strain, G is the shear modulus, and εv is the volu-metric strain.

As the axial boundary is �xed, the axial strain is 0. �us,εv � εr + εθ. �e geometric equations can be written asfollows:

εr � −zurzr,

εθ � −urr,

(6)

where ur is the radial displacement.Because the whole model is axisymmetric, the circum-

ferential displacement uθ is 0. Finally, the stress �eldequations can be written as

E(1− ])(1 + ]) · (1− 2])

z2urzr2

+1r

zurzr−1r2ur( )− β

zp

zr� 0. (7)

3.2. Seepage Field. According to the de�nition of porosity,the porosity φ can be written as

φ �Vp

Vb�Vp0 + ΔVp

Vb0 + ΔVb� 1−

Vs0 + ΔVs

Vb0 + ΔVb

� 1−Vs0 1 + ΔVs/Vs0( )( )Vb0 1 + ΔVb/Vb0( )( )

� 1−1−φ0( ) 1 + ΔVs/Vs0( )( )

1− εv,

(8)

7

6

5

4

3

2

1

00.00 0.05 0.10 0.15 0.20 0.25

Displacement (mm)

Load

(kN

)

0.30 0.35 0.40

BT-1BT-2

BT-3BT-4

Figure 7: Load-displacement curves of the �attened Brazilian diskspecimens. BT-1, BT-2, BT-3, and BT-4 denote the Brazilian disctest of specimens 1, 2, 3, and 4, respectively.

Table 1: Tensile strength and fracture toughness of the redsandstone specimens.

Testingnumber

D(mm)

R(mm)

H(mm)

Pc(kN)

Pmin(kN)

σt(MPa)

KIC(MPa·m1/2)

1 49.98 24.99 16.18 6.76 3.91 5.11 1.222 49.92 24.96 17.70 6.34 3.20 4.39 0.913 49.86 24.93 16.88 5.77 3.49 4.19 1.044 50.01 25.05 15.94 5.62 2.53 4.30 0.80

6 Advances in Civil Engineering

8

7

6

5

4

3

2

1

0250 300

Pres

sure

(MPa

)

350 400Time (s)

Pressure

Volumetric strain

Volu

met

ric st

rain

450 500 550

–0.004

–0.003

–0.002

–0.001

0.000

0.001

(a)

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.00 20 40 60 80 100 120 140 160 180 200

–0.06

–0.05

–0.04

–0.03

–0.02

–0.01

0.00

Pres

sure

(MPa

)

Time (s)

Volu

met

ric st

rain

Pressure

Volumetric strain

(b)

8

7

6

5

4

3

2

1

0

Pres

sure

(MPa

)

Pressure

Volumetric strain

250 300 350 400Time (s)

450 550500 650600

Volu

met

ric st

rain

–0.005

–0.006

–0.004

–0.003

–0.002

–0.001

0.000

0.001

0.002

(c)

4.04.55.0

3.53.02.52.01.51.00.50.0

Pres

sure

(MPa

)

0 25 50 75 100 125 150 175 200 225 250Time (s)

–0.014

–0.012

–0.010

–0.008

–0.006

–0.004

–0.002

0.000

Volu

met

ric st

rain

Pressure

Volumetric strain

(d)

8

7

6

5

4

3

2

1

0

Pres

sure

(MPa

)

Pressure

Volumetric strain

250 300 350 400Time (s)

450 550500 650600

Volu

met

ric st

rain

–0.004

–0.003

–0.002

–0.001

0.000

0.001

(e)

5

4

3

2

1

0

Pres

sure

(MPa

)

0 50 100 150 200 250 300 350 400 450Time (s)

–0.035

–0.030

–0.025

–0.020

–0.015

–0.010

–0.005

0.005

0.000

Volu

met

ric st

rain

Pressure

Volumetric strain

(f )

Figure 8: Plots of volumetric strain and pressure with time during fracturing. (a) W1 loading rate: 0.09552MPa/s. (b) G1 loading rate:0.03259MPa/s. (c) W2 loading rate: 0.07328MPa/s. (d) G2 loading rate: 0.02254MPa/s. (e) W3 loading rate: 0.06368MPa/s. (f) G3 loadingrate: 0.01237MPa/s.

Table 2: �e volumetric strain of the specimens under the waterfracturing pressure (4MPa).

Specimen number Loading rate (MPa/s) Volumetric strainW1 0.09552 −0.00020W2 0.07328 −0.00017W3 0.06368 −0.00018

Table 3:�e volumetric strain of the specimens under the nitrogengas fracturing pressure (4MPa).

Specimen number Loading rate (MPa/s) Volumetric strainG1 0.03259 −0.0010G2 0.02254 −0.0013G3 0.01237 −0.0017

Advances in Civil Engineering 7

where Vp, Vp0, and ΔVp are the pore volume, initial porevolume, and the increment of the pore volume; Vb, Vb0, andΔVb are the appearance volume, initial appearance volume,and the increment of appearance volume; Vs0 and ΔVs arethe volume of initial solid skeleton in porous media and theincrement of solid skeleton; φ0 � Vp0/Vb0 is the initial po-rosity; and εv � −ΔVb/Vb0 is the volumetric strain.

It is assumed that the deformation of solid skeleton iscaused only by the pore stress:

ΔVs

Vs0�−ΔpKs

. (9)

where Δp is the increment of pore stress.

Main cracks

(a)

Main cracks

Secondarycracks

(b)

Figure 9: Failure patterns after (a) the water fracturing and (b) the nitrogen gas fracturing.

Seepage field-2: P2

Seepage field-1: P1

Stress field: F

13

2

4

r0

r0 r1

r0 r1

Figure 10: Computation routine.

Stress field

Boundary conditions of seepage field

Displacement field

Whether fracture

Yes

No

Output fracture pressure and seepage range

Permeability and porosityInitial pore pressure

Seepage field

Figure 11: �e �ow chart of simulation.

Figure 12: Calculation model of the stress �eld.

8 Advances in Civil Engineering

�e permeability K can be written according to theKozeny–Carman equation [24] as

K �φ3

cS2, (10)

where c is the dimensionless coe§cient which can be give anapproximate value of 5 [24] and S is the speci�c surface areaof the solid phase.

Since the change of the surface area is very small, thepermeability K can be written based on the initial perme-ability K0 as [25]

K �K0

1− εv( )1−

εv − Δp 1−φ0( )/Ks)(φ0

[ ]3

. (11)

Based on the conservation of mass, ∇[ρs(1−φ)V→

s] +z[ρs(1−φ)]/zt � 0, where ρs means the density of rock solidskeleton and V

→s means the absolute speed of rock solid

skeleton particles. Considering that (1−φ)ρs≫ ρs|V→

s|≫(1−φ)|V

→s|, the continuity equation of rock solid skeleton

particles can be written as

(1−φ)ρs∇V→

s +(1−φ)zρszt− ρs

zφzt� 0. (12)

As the same as rock solid skeleton particles, the conti-nuity equation of �uid can be simpli�ed as

φρf∇V→

f + φzρfzt+ ρf

zφzt� 0, (13)

where ρf is the density of �uid or gas and V→

f is the absolutespeed of �uid or gas.

On the contrary, gas can be compressed easily, so thecontinuity equation of gas is

φV→

f∇ρf + φρf∇V→

f + φzρfzt+ ρf

zφzt� 0. (14)

Finally, the seepage �eld equation of �uid under �uid-solid coupling described by pore pressure p is

zεvzt−1μ∇(K∇p) +

K

r

zp

zr[ ] +

1−φKs

+φKf

( )zp

zt� 0. (15)

�e seepage �eld equation of gas is

−K

μpzp

zr( )

2

+φp

zurzt

zp

zr+zεvzt−1μ∇(K∇p) +

K

r

zp

zr[ ]

+1−φKs

+φp

( )zp

zt� 0.

(16)

3.3. Fracture Criterion. �e stress intensity factor isemployed to evaluate whether fracture happens or not asfollows:

K1 +K2 +K3 ≥KIC. (17)

�e whole stress intensity factor of breakdown pressureis divided into three parts: K1 is the stress intensity factor of

in situ stress; K2 is the stress intensity factor of boreholestress; and K3 is the stress intensity factor in pore stress.

�e stress intensity factor can be obtained as follows(shown in Figure 13):

K1 � −0.9872σ���πa

√,

K2 � 0.205f(t)���πa

√,

K3 � ∫r1

r0

p(r, t)���πa

√����a + ra− r

√+����a− ra + r

√( )dr,

(18)

where σ is the vertical stress; α is the lateral stress ratio; a isthe pore length, f(t) is the pressure of the borehole thatchanges with time; p(r, t) is the pore stress that changeswith time and position; and r0 and r1 are the radius of theborehole and the in�uenced radius of pore pressure.

3.4. Results and Analysis. In order to verify the fracturecriterion in the water fracturing and the nitrogen gasfracturing and the accuracy of the calculation method,COMSOL Multiphysics is used to simulate the water frac-turing and the nitrogen gas fracturing tests of the redsandstone under di erent loading rates. �e relevant pa-rameters are listed in Table 4 [26, 27].

3.4.1. Comparison Study of Numerical Modeling andExperiments. �e computational and experimental resultsare listed in Tables 5 and 6. It can be concluded that thevalues of the rupture time and breakdown pressure calcu-lated by numerical approach are close to those obtainedfrom experiments. �erefore, the computational model canbe taken as a reliable tool for predicting the breakdownpressure and the rupture time.

Based on the numerical calculation, the pore pressure in theprocess of the water fracturing and the nitrogen gas fracturingcan be obtained under di erent fracturing pressures and dif-ferent loading rates. �e distribution curves of the porepressure during the fracturing process are plotted in Figure 14.

Figure 14 shows that the pore pressure distributioncurves during water fracturing and the nitrogen gas frac-turing are di erent. �e breakdown pressure of the waterfracturing is higher than that of nitrogen gas fracturing. �e

p (r, t)

f (t)

ασ ασ

σ

σ

a

Figure 13: Calculation model of the stress intensity factor.

Advances in Civil Engineering 9

nitrogen gas has penetrated through the specimen from theradial direction in 40 s; however, the maximum penetrationlength is only about 0.012m under the water fracturing.

'e difference mentioned above may be caused by thedifferent permeation capacity of the water and the nitrogen gas.As we know, compared with the water, the nitrogen gas has theproperties of lower viscosity. 'e nitrogen gas can quicklypenetrate through the specimen from the radial direction. Onthe contrary, water can only permeate to a part of the specimenfrom the radial direction. 'us, for the nitrogen gas fracturing,the gradient of pore pressure drop is lower, and the stressintensity factor is higher than that under the water fracturing ata certain the injection pressure. 'erefore, the values ofbreakdown pressure under the nitrogen gas fracturing arelower than those under the water fracturing. 'is is alsoconsistent with the data obtained in the experiments.

Figure 15 shows that, for the different loading rates of thewater fracturing, the seepage ranges are, respectively, 11.6mm,12.2mm, and 12.5mm. 'e difference of the seepage rangesunder the maximum loading rate and the minimum loading

rate is only 0.9mm.'erefore, the pore pressure distribution ateach breakdown pressure of different loading rates is similar,and the values of breakdown pressure under each loading rateare close. On the contrary, for the nitrogen gas fracturing, thenitrogen gas can easily penetrate the specimen from the radialdirection. Once the nitrogen gas penetrates through thespecimen, the seepage boundary can be taken as fixed.'us, thepore pressure distribution changes little with the loading rate.'e loading rate only influences the rupture time. 'e higherloading rate leads to the shorter rupture time.

'e seepage rate can be determined by the gradient of thepenetration range-time curve (Figure 15). 'e values ofseepage rate are listed in Table 7. It can be observed that, foreither the water fracturing or the nitrogen gas fracturing, theseepage rate increases with the loading rate.

3.4.2. Study on Loading Rate Effect. Based on the results ofSection 3.4.1, the loading rate has little effect on thebreakdown pressure for both the water fracturing and the

Table 4: Calculation parameters.

Physical quantity Unit ValueDrilling radius m 0.005Sample radius m 0.025Young’s modulus GPa 5.6Poisson ratio 0.1Biot’s coefficient 0.9Compressive volumetric modulus of skeleton GPa 200Compressed volumetric modulus of water GPa 2.2Initial porosity 0.1Initial permeability m2 1e− 16Initial pore pressure MPa 0.101Crack length m 0.013Fracture toughness MPa m1/2 0.99Viscosity of water Pa s 1e− 3Viscosity of nitrogen Pa s 0.178e− 4Loading rates of the water fracturing MPa/s 0.09552 (W1), 0.07328 (W2), 0.06368(W3)Loading rates of the nitrogen gas fracturing MPa/s 0.03259 (G1), 0.02254 (G2), 0.01237 (G3)

Table 5: 'e numerical and experimental values of rupture time and breakdown pressure (water fracturing).

Specimen number Method Rupture time (s) Breakdown pressure (MPa)

W1 Numerical modeling 76 7.36Experiment 82.4 7.52

W2 Numerical modeling 96 7.14Experiment 102.2 7.38

W3 Numerical modeling 109 7.04Experiment 119.9 7.75

Table 6: 'e numerical and experimental values of rupture time and breakdown pressure (nitrogen gas fracturing).

Specimen number Method Rupture time (s) Breakdown pressure (MPa)

G1 Numerical modeling 140.0 4.664Experiment 153.2 4.0

G2 Numerical modeling 202.4 4.663Experiment 226.5 4.7

G3 Numerical modeling 374.6 4.662Experiment 389.6 4.2

10 Advances in Civil Engineering

8

7

6

Pres

sure

(MPa

)

5

4

3

2

1

00.005 0.006 0.007 0.008 0.009

R (m)

20 s40 s

60 sBreakage

0.010 0.011 0.012

(a)

Pres

sure

(MPa

)

5

4

3

2

1

00.0050 0.0075 0.0100 0.0125 0.0150

R (m)

20 s40 s

80 s60 s

0.0175 0.02250.0200 0.0250

100 s120 sBreakage

(b)

8

7

6

Pres

sure

(MPa

)

5

4

3

2

1

00.005 0.006 0.007 0.008 0.009

R (m)0.010 0.011 0.012

20 s40 s60 s

80 sBreakage

(c)

Pres

sure

(MPa

)5

4

3

2

1

00.0050 0.0075 0.0100 0.0125 0.0150

R (m)0.0175 0.02250.0200 0.0250

20 s40 s60 s

80 s100 s120 s

140 s160 s180 s

200 sBreakage

(d)

8

7

6

Pres

sure

(MPa

)

5

4

3

2

1

00.005 0.006 0.007 0.008 0.009

R (m)0.010 0.011 0.0120.013

20 s40 s60 s

80 s100 sBreakage

(e)

Pres

sure

(MPa

)

5

4

3

2

1

00.0050 0.0075 0.0100 0.0125 0.0150

R (m)0.0175 0.02250.0200 0.0250

20 s40 s60 s80 s

100 s120 s140 s160 s

180 s200 s220 s240 s

260 s280 s300 s320 s

340 s360 sBreakage

(f )

Figure 14: Plots of pore pressure distribution during fracturing (I). (a) W1, (b) G1, (c) W2, (d) G2, (e) W3, and (f) G3.

Advances in Civil Engineering 11

nitrogen gas fracturing under our experimental condition.�e reason may be that (1) the loading rate does not havea signi�cant e ect on seepage range for the water fracturing;(2) for the nitrogen gas fracturing, the seepage range isconstant as the nitrogen gas can penetrate the specimeneasily, and the seepage boundary can be taken as �xed.

However, the loading rates for the water fracturing andthe nitrogen gas fracturing are not consistent in Section2.4.2; for example, the loading rates of W1 and G1 aredi erent. As mentioned in Section 2.3, the loading rates forthe water fracturing and the nitrogen gas fracturing canhardly be the same because of the equipment condition.Meanwhile, the numerical method is validated to be reliablein Section 3.4.1. �erefore, the loading rate e ect is in-vestigated in this section by using numerical modeling, withthe same loading rate for the water fracturing and the ni-trogen gas fracturing.

Moreover, for the water fracturing, the seepage rangechanges with the loading rate although the di erence is notobvious. It implies that the loading rate may a ect theseepage range. If the loading rate varies greatly in the waterfracturing, the seepage range may vary greatly, and theloading rate may have a signi�cant e ect on the breakdownpressure. �erefore, three loading rates, 0.5MPa/s (W1′,G1′), 0.05MPa/s (W2′, G2′), and 0.005MPa/s (W3′, G3′),are selected to investigate the loading rate e ect.

�e values of rupture time and breakdown pressureare summarized in Tables 8 and 9. For the water fractur-ing, the breakdown pressures of the water fracturing are8.60MPa, 6.80MPa, and 5.36MPa under the loading rates of0.5MPa/s, 0.05MPa/s, and 0.005MPa/s. A higher loadingrate results in a higher breakdown pressure. And the rupturetime increases from 15.8 s to 1053 s when the loading ratedecreases from 0.5MPa/s to 0.005MPa/s. When the loadingrate varies greatly, the seepage range varies greatly. �eloading rate has a signi�cant e ect on the breakdownpressure. Meanwhile, from Figures 16 and 17, it can beobserved that the loading rate of the water fracturing variesgreatly, and the seepage range extends obviously. �e

W1W2W3

0 10 20 30 5040 60Time (s)

70 9080 110100 120

0.013

0.012

0.011

0.010

0.007

0.008

0.006

0.009

0.005

Pene

trat

ion

rang

e (m

)

(a)

0.030

0.025

0.020

0.015

0.010

0.0050 50 100 150 200

Time (s)250 300 350

G1G2G3

400

Pene

trat

ion

rang

e (m

)

(b)

Figure 15: Plots of the seepage range with time during fracturing (I). (a) Water fracturing. (b) Nitrogen gas fracturing.

Table 7: Results of seepage rate (I).

Specimen number Method Seepage rate (mm/s)W1 Numerical modeling 0.0786W2 Numerical modeling 0.0703W3 Numerical modeling 0.0659G1 Numerical modeling 0.7621G2 Numerical modeling 0.6068G3 Numerical modeling 0.4223

Table 8: �e numerical results of rupture time and breakdownpressure with the water fracturing.

Specimennumber Method Rupture

time (s)Breakdown

pressure (MPa)

W1′ Numericalmodeling 15.8 8.6

W2′ Numericalmodeling 135 6.8

W3′ Numericalmodeling 1053 5.36

Table 9: �e numerical results of rupture time and breakdownpressure with the nitrogen gas fracturing.

Specimennumber Method Rupture

time (s)Breakdown

pressure (MPa)

G1′ Numericalmodeling 9.6 4.80

G2′ Numericalmodeling 91.6 4.67

G3′ Numericalmodeling 911.8 4.66

12 Advances in Civil Engineering

8

7

6

5

Pres

sure

(MPa

)

4

3

2

1

00.0050 0.0075 0.0100 0.0125 0.0150

R (m)0.0175 0.0200 0.0225 0.0250

2 s4 s6 s

8 s10 s12 s

14 sBreakage

(a)

5

Pres

sure

(MPa

)

4

3

2

2 s4 s6 s

8 sBreakage

1

00.0050 0.0075 0.0100 0.0125 0.0150

R (m)0.0175 0.0200 0.0225 0.0250

(b)

20 s40 s60 s

80 s100 s

120 sBreakage

8

7

6

5

Pres

sure

(MPa

)

4

3

2

1

00.0050 0.0075 0.0100 0.0125 0.0150

R (m)0.0175 0.0200 0.0225 0.0250

(c)

20 s40 s60 s

80 sBreakage

5

Pres

sure

(MPa

)4

3

2

1

00.0050 0.0075 0.0100 0.0125 0.0150

R (m)0.0175 0.0200 0.0225 0.0250

(d)

200 s400 s600 s

800 s1000 sBreakage

8

7

6

5

Pres

sure

(MPa

)

4

3

2

1

00.0050 0.0075 0.0100 0.0125 0.0150

R (m)0.0175 0.0200 0.0225 0.0250

(e)

200 s400 s600 s

800 sBreakage

5

Pres

sure

(MPa

)

4

3

2

1

00.0050 0.0075 0.0100 0.0125 0.0150

R (m)0.0175 0.0200 0.0225 0.0250

(f)

Figure 16: Plots of pore pressure distribution during fracturing (II). (a) W1′, (b) G1′, (c) W2′, (d) G2′, (e) W3′, and (f) G3′.

Advances in Civil Engineering 13

maximum penetration distance of W1′, W2′, and W3′ are8.7mm, 13.1mm, and 23.1mm, respectively. �erefore, thepore pressure distribution varies obviously, and the gradientof pressure drop, which causes the change of breakdownpressure, varies signi�cantly.

However, for the nitrogen gas fracturing, the nitrogengas can penetrate through the specimen from the radialdirection under each loading rate. Hence, the breakdownpressure of the water fracturing and the seepage range do notchange obviously. �e breakdown pressures under di erentloading rates of the nitrogen gas fracturing are all about4.70MPa.

Similar to Table 7, the values of the seepage rate underthe three loading rates (0.5MPa/s, 0.05MPa/s, and0.005MPa/s) are listed in Table 10. It can be obtained thatthe seepage rate increases obviously with the loading rate forboth the water fracturing and nitrogen gas fracturing.

4. Conclusions

In this study, the fracturing behavior of the red sandstoneunder water fracturing and the nitrogen gas fracturing areinvestigated by experiments and numerical modeling. �econclusions can be summarized as follows:

(1) Based on the fracturing tests of the red sandstonewith water and nitrogen gas, it can be found that

breakdown pressure of the water fracturing is higherthan the one under the nitrogen gas fracturing. �ebreakdown pressure of the nitrogen gas fracturing isabout 60% that of the water fracturing.

(2) �e reason of the phenomenon mentioned above isthat the nitrogen gas has the property of ultralowviscosity. �e nitrogen gas can transmit the injectionpressure to all the penetrated points, and the porepressure is higher.�us, a lower breakdown pressureis needed to breakdown the specimen.

(3) �e volumetric strain of the specimen in nitrogen gasfracturing is larger than that in hydraulic fracturing.Meanwhile, the crack network formed under thenitrogen gas fracturing is more su§cient than thatunder the water fracturing.

(4) �e breakdown pressure and rupture time of the waterfracturing and the nitrogen gas fracturing calculatedby the numerical method are close to those obtainedfrom the experimental data. It implies that the nu-merical method can be a reliable tool for predicting thebreakdown pressure and the rupture time.

(5) To investigate the loading rate e ect on the break-down pressure and the rupture time, a sensitive studybased on three loading rates (0.5MPa/s, 0.05MPa/s,and 0.005MPa/s) is carried out by numericalmodeling. For the water fracturing, the loading ratewill in�uence the seepage range; therefore, it willhave a signi�cant e ect on pore pressure distributionand the breakdown pressure. A higher loading rateresults in a higher breakdown pressure. On thecontrary, for the nitrogen gas fracturing, the nitrogengas can easily penetrate the specimen from the radialdirection; thus, the seepage boundary can be taken as�xed. �e breakdown pressure almost does not

W1′W2′W3′

Pene

trat

ion

rang

e (m

)

0.025

0.020

0.015

0.010

0.005120010008006004002000

Time (s)

(a)

G1′G2′G3′

0.030

10008006004002000

Pene

trat

ion

rang

e (m

)

0.025

0.020

0.015

0.010

0.005

Time (s)

(b)

Figure 17: Plots of the seepage range with time during fracturing (II). (a) Water fracturing. (b) Nitrogen gas fracturing.

Table 10: Results of the seepage rate (II).

Specimen number Method Seepage rate (mm/s)W1′ Numerical modeling 0.1877W2′ Numerical modeling 0.0596W3′ Numerical modeling 0.0169G1′ Numerical modeling 3.8462G2′ Numerical modeling 0.9921G3′ Numerical modeling 0.2427

14 Advances in Civil Engineering

change with the loading rate. And a higher loadingrate results in a shorter rupture time and a higherseepage rate.

Data Availability

Data supporting this research article are available from thecorresponding author on request.

Conflicts of Interest

'e authors declare that they have no conflicts of interest.

Acknowledgments

'is project was supported by the Fundamental ResearchFunds for the Central Universities (no. 2018QNA33). 'eauthors acknowledge the support of the China University ofMining and Technology.

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