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Journal of Natural Gas Science and Engineering 46 (2017) 651e663

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Journal of Natural Gas Science and Engineering

journal homepage: www.elsevier .com/locate/ jngse

Numerical modelling of fracturing effect stimulated by pulsatinghydraulic fracturing in coal seam gas reservoir

Chunchi Ma a, b, Yupeng Jiang b, *, Huilin Xing b, Tianbin Li a

a State Key Laboratory of Geohazard Prevention and Geoenvironment Protection, Chengdu University of Technology, Chengdu 610059, Chinab Centre for Geoscience Computing, School of Earth Sciences, The University of Queensland, St Lucia, QLD 4072, Australia

a r t i c l e i n f o

Article history:Received 20 January 2017Received in revised form4 June 2017Accepted 15 August 2017Available online 7 September 2017

Keywords:Pulsating hydraulic fracturingPermeability enhancementFracturing mechanismNumerical simulationFracturing parameter

* Corresponding author.E-mail address: yupeng.jiang@uq.net.au (Y. Jiang).

http://dx.doi.org/10.1016/j.jngse.2017.08.0161875-5100/© 2017 Elsevier B.V. All rights reserved.

a b s t r a c t

Pulsating hydraulic fracturing (PHF) technology is an advanced permeability enhancement method forcoal seam gas mining. Laboratory and field experiments indicate that PHF can stimulate a well-distributed fracture system inside a coal reservoir. However, the basic mechanism behind this effect isstill poorly understood. In this study, a better mathematical model for pressure ripple propagation isproposed and an analytical solution is obtained. Furthermore, the particle flow code is applied based onthe analytical solution to numerically simulate the fracturing effect of PHF. The mechanism for fracturesystem formation with the original coal cleat system is quantitatively analysed by using advanced in-dicators (crack event density, crack intensity rate and kinetic energy). A new cracking pattern is proposedand discussed. Eventually, fracturing effects under different engineering PHF inputs (i.e., pulsating fre-quency and ripple amplitude) are numerically simulated and analysed. The conclusions build a theo-retical basis for the mechanism of PHF effect. The PHF parameters may also be largely improved andoptimized for the extension and formation of fracture networks in a coal seam gas reservoir.

© 2017 Elsevier B.V. All rights reserved.

1. Introduction

Hydraulic fracturing (HF) is a major mining technology for tightformation reservoirs (Al Rbeawi and Tiab, 2013; Gandossi, 2013)and has also been applied for coal permeability enhancement(Karacan et al., 2011). Fracture networks stimulated by HF mainlycontain two types of fractures: main fractures and branch fractures.As the passageways of gas flow and diffuse (Li and Xie, 2004), well-distributed fracture networks, especially for branch fractures, canhighly increase the permeability of a coal seam gas reservoir. Due tothe fragility of coal, however, fracturing effects of traditional hy-draulic fracturing methods are highly compromised (Jiang andXing, 2016). Under relatively stable and high fracturing pressures,which are commonly used in HF, branch fractures do not haveenough time to be fully developed (Huang et al., 2011). Hence, it isimperative to develop an advanced fracturing technology to formwell-distributed, fully developed branch fracture networks in coalreservoirs.

In the 1980s, a new concept of pulse loading was initiallyintroduced in HF process to improve the fracturing effect in the oil

industry. By using this pulse loading method (e.g. explosion ofcharge located in the well or operation of submerged powder gas-generator) more and longer cracks could be produced (Sher andAleksandrova, 2002; Wang, 1987). In order to make the pulseloading more applicable, the controlled pulse fracturing (CPF) wasproposed to achieve the gradual pulse increase in thewell pressure.In field tests, multiple radial vertical fractures are created byimplementing CPF in wellbores, which effectively expand thefracture system (Uhri and Prairie, 1988). Lately, pulsating hydraulicfracturing (PHF) technology has been developed as an effectivemethod to enhance permeability in a weak formation reservoir,especially for coal seam gas reservoirs. The fundamental idea of PHFis to create fluid pressure ripples by injecting fracturing fluid with acertain pulsating frequency. Under this low periodical pressureloading, a well-distributed fracture network can be produced withfully extended branch fractures and the rupturing progress of coalmass can be accelerated. A few laboratory studies have been con-ducted to seek the PHF mechanism. A variable frequency pulsehydraulic fracturing testing systemwas built (Li et al., 2013) and theinfluence of pulse frequencies for fracturing effects was studied (Liet al., 2014). The pulsating water pressure propagation in coalfractures during PHF was experimentally studied by using asimplified model (Zhai et al., 2015).

List of symbols:

BPM Bonded particle modelCED Crack event densityCIR Crack intensity rateDFN Discrete fracture networkKE Kinetic energyPHF Pulsating hydraulic fracturingSJ Smooth joint modelSRM Synthetic rock mass model

C. Ma et al. / Journal of Natural Gas Science and Engineering 46 (2017) 651e663652

Due to the complexity of field conditions (geological structure,crustal stresses etc.), attempts to approximate the field PHF effectsin laboratory studies are restricted. The basic mechanisms forpressure ripple propagation and formation of a fracture networkare still poorly investigated. Particularly, the impacts of the originalcoal cleat system for fracturing effects have not been considered inexperiments because of the existing experimental limitations.Therefore, numerical simulation is introduced as an alternative andeffective way to study the basic mechanism of the PHF fracturingeffect. Current numerical schemes for HF cannot be directly used inPHF simulation due to the following two reasons: (1) commonnumerical methods (e.g. FEM and FDM) for HF simulation (Guoet al., 2015; Zhang and Bian, 2015; Zhao et al., 2014; Zhou andHow, 2013) are not suitable (at least not directly) for the simula-tion of PHF. The details are further discussed in the later part of thispaper (2) pressure ripples after propagation in the fracturing pipeare different from the input pressures. An accurate analytical so-lution for pressure ripples is required to approximate the actualloading condition of PHF.

To solve these problems proposed above, an advanced mathe-matical solution for the propagation of pressure ripples is proposed.Based on the analytical solution, numerical simulation is conductedby using the three-dimensional particle flow code, which couldwell characterize the formation, development, and connection ofmicro-cracks, and the stress distribution affected by the interactionof fractures. The original cleat system and the contacting relation-ships are fully considered in the coal model; tri-axial loading isintroduced to simulate crustal stresses. Fracturing effects underdifferent input schemes and parameters (i.e. pulsating frequencyand pressure amplitude) are simulated for engineering optimiza-tion. The simulation results are expected to provide a valid expla-nation for the fracturing mechanism of PHF.

2. Pulsating pressure and propagation

PHF technology uses a piston pump to create sinusoidal waterpressure at the entrance of a fracturing pipe. This pressure changecan propagate inside the pipe and generate pressure ripples(Bergada et al., 2012; Harrison and Edge, 2000). Compared with HF,the sinusoidal pressure has relatively lower value withmore drasticpressure changes (Li et al., 2014; Zhai et al., 2015). Unlike thetraditional fracturingmethod, which relies on seepage to propagatepressure, the PHF allows fracturing pressures to propagate in a formof mechanical wave with a velocity that much higher than thevelocity of seepage. Hence the changing pattern of pulsatingpressure ripples at the end of a fracturing pipe has a major impacton fracturing effects. It is imperative to give the valid mathematicalsolution for engineering optimization. Many mathematical modelshave been proposed to describe water pressure changes in narrowfractures. The percolation model (Rehbinder, 1977) and transient

flow model (Fiorotto and Rinaldo, 1992) are the most acknowl-edged models. These models are effective approaches for pressurepropagation in dam fractures and long water pipes, both consid-ering the velocity of water flow inside the fracture pressure (frac-turing pipe is unsaturated and has relatively large physical scales);the high-frequency vibrations that are generated from inherentfrequencies can also be numerically calculated from these partialdifferential equations (PDE). Regarding PHF, however, these modelsare not appropriate for describing the patterns of pressure propa-gation. Under PHF stimulation, a fracturing pipe is filled withfracturing fluid within a short pulsating time and the fracturingfluid is confined in an extremely narrow space with high pressure.Meanwhile, PHF uses low frequencies (e.g., 15 Hz, 20 Hz, 25 Hz,etc.), which make the wavelengths of pressure ripples 102 to 103

times larger than the geometrical scale of coal fractures. Hence, thefracturing fluid flows for pressure propagation are negligible andshould be eliminated from the mathematical model.

2.1. Analytical solutions for pressure propagation

A straight pipe filled with water should be simplified as a one-dimensional mode. By solving the elastic wave equation in onedimension, the analytical solution for a water-filled pipe's longi-tudinal vibration can be accurately obtained. A uniform mode isconsidered to represent the longitudinal vibration of a straightwater-filled pipe. The governing PDE can be written as

rwv2uvt2

� lv2uvx2

¼ 0; 0 � x � L; (1)

where u(x, t) is the longitudinal displacement of the water pipe atdistance x and time t. rw, l, and L are the density, bulk modulus offluid and length of pipe, respectively. The boundary conditions andinitial conditions can be written as

vuvx

����x¼0

¼ f ðtÞ ¼ 1lðB0 þ A0 sinðutÞÞ; uðL; tÞ ¼ 0;

vuvt

����t¼0

¼ 0; uðx;0Þ ¼ 0 (2)

Physically, boundary conditions represent a free-clamped waterpipe. The fluid entrance is set as a free boundary for pressure input,with coefficient A0, constant parameter B0 and pulsating angularfrequency u. The end of a fracturing pipe is set as a zero displace-ment boundary because the PDE discusses the pressure propaga-tion prior to fracture extension. The solution for equation (1) isassumed as

uðx; tÞ ¼ U1ðx; tÞ þ U2ðx; tÞ; (3)

where U1(x, t) is the general solution and U2(x, t) represents aparticular solution. The general solution can be obtained by usingthe method of separation of variables, and after reducing theboundary conditions to homogeneous conditions, the general so-lution can be written as

U1ðx; tÞ ¼X∞n¼1

½Cn sinðuatÞ þ Dn cosðuatÞ�cos�ubx

�; (4)

ua ¼ ð2n� 1Þpa2L

; ub ¼ ð2n� 1Þp2L

; (5)

where a denotes the velocity of pressure propagation in the water-filled pipe model; Cn and Dn are the coefficients for the solution

C. Ma et al. / Journal of Natural Gas Science and Engineering 46 (2017) 651e663 653

U1(x, t), which are related to initial conditions calculated from theparticular solution.

Cn ¼ 2abL

ZL

0

���vU2

vt

����t¼0

�cos

�ubx

�dxz

16A0uL2

a½ð2n� 1Þp�3(6)

Dn ¼ 2L

ZL

0

ð � U2ðx;0ÞÞcos�ubx

�dx ¼ 8B0L

½ð2n� 1Þp�2(7)

Similarly, the particular solution in equation (9) can be writtenas

U2ðx; tÞ ¼ B0ðx� LÞ þ A0 sinðutÞ�ausin

�uax�

� autan

uLa

cos�uax��

:

(8)

The pressure changes can be calculated by solving thedisplacement equation:

Pðx; tÞ ¼ ldudx

: (9)

2.2. Solution analysis

The fully saturated pipe model is proposed so that the high-frequency components of the pressure ripple in a fracturing pipe,which are observed from experiments and field tests (Li et al., 2014;Wang et al., 2014, 2015), can be accurately obtained from this so-lution. Fig. 1 shows the non-dimensionalized input pressure andripples at the end of a fracturing pipe. The dashed line denotes theinput sine function f(t), with ripple amplitude A0 and parameter B0.The specific values of both parameters are decided through thedesignation of the piston pump of the pulsating machine. The solidline represents the analytical solution of pressure ripple propaga-tion, using f(t) as the pressure boundary condition. Mathematically,the high-frequency components are included in general solution U1,while the coefficients Cn and Dn are determined by the generalsolution and geometrical parameters of the pipe model (i.e.,equation (6) and equation (7)), respectively.

From a physical perspective, ua represents the inherent fre-quency of thewater pipe, whichmeans that the vibrating responsesfor pressure ripples can only possess specific frequencies. Theseparameters are determined by the physical properties and thelength of the water-filled pipe. The integer n denotes the scale ofthe inherent frequency; the frequency is higher with larger scales.Under continuous pulsating, pressure ripples propagate throughthe fluid and reflect back and forth in the pipe. Ripples cannotsimplymaintain their original sinusoidal shape but rather graduallyform a standing wave inside the pipe. The frequency of thisstanding wave is assembled by a series of harmonic waves withdifferent inherent frequencies and the input pressure. Hence, thehigh-frequency components are generated. In this research, thepressure ripple calculated from the analytical solution is used fornumerical simulations.

2.3. Limitation of the solution

The boundary conditions are valid for these PHF laboratory ex-periments. Because they are conducted in a small coal sample thatseal by steel plates to load the surrounding stress (Li et al., 2014,

2013); or in a sealed steel pipe (Zhai et al., 2015). In these cases,the wave energy is largely confined by strong impedance (e.g.steel), which can be approximated by a boundary condition of azero displacement. By using the Zoeppritz equations, the propor-tion of wave energy is calculated. The equation can be written as

2666666666664

sina cosb �sina0

cosb0

cosa �sinb cosa0

sinb0

sin2aVp1

Vs1cos2b

Vp1

Vp2

V2s1

V2s2

r2r1

sin2a0 �r2

r1

Vp1Vs2

V2s1

cos2b0

cos2b �Vp1

Vs1sin2b �r2

r1

Vp2

Vp1cos2b

0 �r2r1

Vs2

Vp1sin2b

0

3777777777775

�

2664rpprpstpptps

3775

¼

2664

�sinacosasin2a�cos2b

3775;

(10)

where a is the incident and reflected P wave angle (rpp), b is thereflected S wave angle (rps), a

0is the refracted P wave angle (tpp),

and b0is refracted S wave angle (tps). The subscribe ‘P’ and ‘S’ denote

the wave type; the ‘1’ and ‘2’ represent the different medium. Therpp, rps, tpp and tps is the rate of displacement amplitude versusincident pressure amplitude. For the PHF, the water pressure waveis P wave with a zero incident angle, so the all wave angles equal tozero. Fig. 2 exhibits the wave energy rate with five different waveimpedance based on Equation (10).

It can be observed that nearly 90% of wave energy will be re-flected back by water/steel impedance and 80% for the coal/steelimpedance. Hence the boundary condition with a zero displace-ment is a valid approximation, and the solution proposed in thispaper is good for the explanation and numerical simulation PHFlaboratory results.

Meanwhile, it must be pointed out that water/coal impedancecannot be treated as zero displacement; because nearly 90% of thewave energy propagates into coal medium and only 10% of thewave energy can be reflected back. Therefore, in real coal reservoir,the high frequency cannot be naturally generated; so the fracturingeffect may not be as good as that of the laboratory. Nearly all of theliterature fail to take this energy dispersion effect intoconsideration.

However, since the latter part of this paper indicates that abetter fracturing effect can be achieved by pressure wave with highfrequency, a ‘seal-release’ pulsating scheme should be used toimprove the PHF technology. The scheme means to seal the frac-turing pipe with steel for a certain time to generate high-frequencycomponents, and then release the new pressure wave form intocoal reservoirs. By using this scheme the high-frequency compo-nents that same with laboratory experiment can be produced.

3. PHF simulation and mechanism interpretation

To properly understand the mechanism of the fracturing effectstimulated by pressure ripples in coal reservoirs (the “coal mass”could be replaced by other mediums by modifying the mechanicalproperties), the interactive relationships between coal properties,coal cleats, and fluid pressures need to be clearly discussed. How-ever, existing experimental apparatuses cannot directly observe

Fig. 1. (a) Input pressure and pressure ripples at the end of the fracturing pipe, (b) enlarged circular area for the high-frequency components.

C. Ma et al. / Journal of Natural Gas Science and Engineering 46 (2017) 651e663654

these phenomena, which is relatively easy to achieve via numericalsimulations. Discrete element methods (e.g. the particle flow code,PFC3D) possess unique advantages for simulations concerned withthe formation, growth, and eventual interaction of micro-cracks(Mas Ivars, 2010). Meanwhile, they can define and build the parti-cle models that fully consider the original coal cleat system. In thispaper, advanced PFC3D model is proposed to research the fracturingmechanism by PHF stimulation. The proposed analytical solution,which includes the high frequency components of fracturingpressures, is adopted as input functions and studied withincomparative situations. Mechanical behaviours (velocities, energy,and cracks) under pressure ripples are accurately measured. Hence,the basic fracturing mechanism for PHF can be properly observed

and discussed.As we discussed early, the finite-element-relatedmethods could

also be applied to simulation the PHF. However, there are majorproblems that make the FEM not as suitable (or practical) as DEM.First of all, the wavelengths are required to fulfil CFL (Courant-Friedrichs-Lewy) condition and grid dispersion condition toconduct a stable simulation, which means in order to simulate thefracturing effect of high-frequency components the number oftemporal steps and grids in FEMneeds to be dramatically increased.It cannot be achieved just by using mesh refinement because thisnumerical stability for wave propagation requires the whole modelto fulfil the CFL condition. Secondly, the DFNmodel, which is crucialfor the simulation of PHF in a coal sample, is not very easy to be

Fig. 2. Wave energy rate for impedance constructed by the different medium; the units for velocity Vp and density Rho are (km/s) and (kg/m3) respectively.

C. Ma et al. / Journal of Natural Gas Science and Engineering 46 (2017) 651e663 655

achieved on the FEM; the highly fractured model easily cause dif-ficulty for the convergence of stiffness matrix. Meanwhile, themesh distortion is also hard to avoid with high-frequencycomponents.

The basic mechanism of PHF is fundamentally different with thetraditional HF. Therefore, for better the calculation efficiency andnumerical stability, the discrete element methods are recom-mended and adopted in this paper.

3.1. Implementation of hydraulic coupling in PFC3D

The hydraulic coupling algorithm is introduced in the programto simulate the hydraulic fracturing effect (ICG, 2008). A com-pacted, bonded assembly of particles is firstly generated. The voidgeometry in an assembly of four neighbouring particles is regardedas being identical to the actual space between particles. A “domain”is defined as a pore created by every four neighbouring particles(Fig. 3a) such that each link in the chain is a side of a tetrahedron (Liand Holt, 2002). Each link (termed flow channel) between twoadjacent domains has a small space between three neighbouringballs (Fig. 3b). As far as fluid is concerned, the flow channel isequivalent to a cylindrical pipe with length Lp and aperture w. Theflow rate (volume per unit time) in a flow channel is given by

Fig. 3. Sketch of Domain-channel model in PFC3D.

q ¼ kw3DPLp

; (11)

where k is a conductivity factor; DP is the pressure difference be-tween two adjacent domains. The length of the flow channel Lp isconsidered to be the distance between the centres of the adjacentdomains in question.

Each domain receives flowsP

q from surrounding channels. Inone time step, i.e., Dt, the increase in hydraulic pressure is given bythe following equation, assuming that inflow is taken as positive:

DP ¼ Kf

VdðSqDt � DVdÞ; (12)

where Kf is the fluid bulk modulus and Vd is the apparent volume ofthe domain. The second term (DVd) represents the mechanicalchange in volume of the domain, i.e., the mechanical changes indomain volumes cause changes in domain pressures.

Each domain accumulates the hydraulic pressure P, which exertstractions on the enclosing particles. The hydraulic pressure in adomain is uniform and the tractions are independent for the patharound a domain. If the polygonal path joins the contacts sur-rounding a domain, the force vector on a typical particle is

Fi ¼ Pnis; (13)

where ni is the unit normal vector of the line joining the centres of adomain and a particle; s is a projective area on the particle in adomain.

Fluid flows and fluid-mechanical coupling effects are calculatedat each time step. Particles are subjected to the laws of motion; therelative motion of two adjacent particles causes overlap or isola-tion, which generates contact forces; local contact forces breakbonds under the cracking criteria from Ma et al. (2016, 2015).

3.2. Establishment of PHF simulation

3.2.1. (1) Coal modelThe synthetic rock mass (SRM) model provides the ability to

conduct numerical experiments by combining two simulation

C. Ma et al. / Journal of Natural Gas Science and Engineering 46 (2017) 651e663656

approaches (Mas Ivars et al., 2008): the Bonded Particle Model(BPM) and the Discrete Fracture Network (DFN). The BPM uses theparticle flow code to assemble and bond particles in three di-mensions (Potyondy and Cundall, 2004). The DFN is inserted intothe BPM by considering the distribution of smooth joint (SJ) con-tacts (Mas Ivars et al., 2011; Mas Ivars, 2010), and thus creating asynthetic coal mass. Fig. 4a shows the SRM coal model for PHFstimulation. More than 64 000 particles (maximum radius 6.5 mm;minimum radius 4.6 mm) are assembled into a cube with thelength 500.0 mm; porosity of the coal model is approximately 12%.Parameters for the model's micro-structures are listed in Table 1.

Three sets of coal cleat are incorporated in the model: beddingplane, face cleat and butt cleat, which are built inside with non-uniform spacing of 50.0 mm, 40.0 mm and 80.0 mm, respectively.The smooth joint contact defines the contact relationship in coalcleats and correlative parameters are exhibited in Table 2. Me-chanical behaviours (e.g. friction and dilatation) on coal cleatsproduce a significant impact on stress distribution, crack behaviourand strength anisotropy of the coal model.

Based on the micro-structural parameters in Table 1 and theparameters for the coal cleat system in Table 2, macro mechanicalproperties of the coal model are approaching real coal samples(referring to a real coal sample with uniaxial compressive strengthsuc ¼ 7.5 MPa, Young's modulus E ¼ 3.5 GPa, and Poisson's ration ¼ 0.48). The mechanical properties of the coal model can bedescribed as follows. Uniaxial loading along the “x” axis gives theproperties: suc ¼ 18.3 MPa, E ¼ 3.8 GPa and n ¼ 0.41; along the “y”axis, suc ¼ 17.1 MPa, E ¼ 3.3 GPa and n ¼ 0.43; along the “z” axis,suc ¼ 18.8 MPa, E ¼ 3.3 GPa and n ¼ 0.41. The anisotropy of macromechanical properties are formed because of the existence of thecoal cleat system.

3.2.2. (2) Pressure schemesPHF simulation is carried out based on the calibrated coal model.

The basic ideas and procedures of numerical simulation arepartially borrowed from Li's PHF experimental system (Li et al.,2014). Fig. 4b shows the sketch of PHF numerical simulation. Sixwalls are set up to confine the coal model as the tri-axial preloadingsystem and energy absorbing boundaries. 10-MPa preloading isapplied to simulate the original crustal stresses in the coal reservoir,which enhances the original mechanical properties of the coalmodel. The hydraulic pressure stimulated by PHF is loaded in a20.0-mm cubic space (fracturing space) at the centre of the coalmodel. A fracturing pipe with the 500-mm length is supposed toextend to the fracturing space and aids to provide the analytical

Fig. 4. Sketch of PHF simu

solution of the input pressure.To stimulate pressure ripples at the centre of the model, the

hydraulic pressure boundary (not the injection rate boundary) isapplied within the fracturing space's domains. Three input schemesof hydraulic pressure are proposed in comparison to seeking themechanism of PHF stimulation (one cycle is illustrated in Fig. 5):

Scheme I is proposed to simulate the traditional HF, with aconstant hydrostatic pressure of 22 MPa.

Scheme II is a sinusoidal pressure ripple (20-Hz frequency, 20-MPa average pressure, and 2-MPa amplitude). It is proposed tostudy the fracturing effect without high-frequency components.

Scheme III is obtained from the analytical solution, by usingscheme II as input function f (t), with ten orders of high-frequencycomponents. It is aiming to study the actual fracturing effect afterpropagation of the sinusoidal pressure ripples along the fracturingpipe. Correlative parameters used to generate scheme III aredetailed in Table 3.

When scheme II or scheme III is applied in themodel, a dynamicsolution (which is the default behaviour of PFC3D) is fully activated.The full dynamic equations of motion including inertial terms aresolved. The generation and dissipation (by the default damping andfrictional behaviours) of kinetic energy are considered in the dy-namic solution. The pressure ripples are implemented at each timestep after the motion calculation via a “Fishcall” function.

To clearly discuss the fracturing effect by pressure ripples, alimited process of seepage (by setting a small flowing step of 10�4 s)is designed to strengthen the mechanical effect of pressure ripples(i.e. dynamic action on the model) and weaken the effect of fluidflows (flow range is limited around the fracturing space). The timeduration for all schemes is 2.8 s (including 56 cycles of pressureripples).

3.3. PHF simulation and validation

The next is to simulate and clarify the fracturing mechanism ofPHF (scheme III) by comparingwith the other two schemes. Variousadvanced indicators are introduced to assess the fracturing effectsand validation of PHF simulation is discussed later.

3.3.1. (1) Crack propagationThe fracturing process is monitored and discussed by various

indicators. The particle velocity stimulated by PHF is fundamentalfor crack propagation. Velocities (located in top, bottom, left andright) around the fracturing space are measured in scheme II and III(velocity fluctuation is negligible in scheme I). Constant vibrations

lation in coal model.

Table 1Calibrated parameters of bond and particle in coal model.

Parameters Rmax

(mm)Rmin

(mm)j r

(kg/m3)Ep(GPa)

Ec(GPa)

knp/ksp kn

c/ksc sc(MPa)

Dsc(MPa)

tc(MPa)

Dtc(MPa)

mp

Values 6.5 4.6 12% 20.5 5.5 6.2 3.5 3.1 20 4 25 3 0.5

Note: Rmax-Maximum particle radius; Rmin-Minimum particle radius; j-Porosity; r-Particle density; Ep-Young's modulus of particle; Ec-Young's modulus of parallel bond; knp/ksp-Normal/Shear stiffness ratio of particle contact; knc/ksc-Normal/Shear stiffness ratio of parallel bond; sc-Tensile strength of bond; tc-Shear strength of bond; Dsc, Dtc-

Standard deviation of tensile strength and shear strength; mp-Frictional coefficient of particle contact.

Table 2Parameters for the three sets of coal cleat.

kn(GPa/m) ks(GPa/m) J(�) mc

Bedding Plane 40 40 10 0.5Face Cleat 30 30 5 0.5Butt Cleat 30 30 6 0.5

Note: kn-Normal stiffness of smooth joint; ks-Shear stiffness of smooth joint; J-Dilatational angle; mc-Frictional coefficient of smooth joint contact.

Table 3Calculation parameters for scheme III.

L(m)

Рw

(kg/m3)l

(GPa)a(km/s)

u

(Rad/s)f(Hz)

T(s)

DPa(MPa)

Paverage(MPa)

N

0.5 1000.0 2.2 1.5 40p 20 2.8 2.0 20.0 20

C. Ma et al. / Journal of Natural Gas Science and Engineering 46 (2017) 651e663 657

of particles inside the coal model can be recorded due to thestimulation of PHF pressure ripples (Fig. 6). These movements canresult in cracking developments. Particle velocities in scheme IIhave a more distinct sinusoidal tendency due to the relativelysimple injecting function, while the counterparts in scheme III arechanging more drastic and irregular because the high-frequencycomponents from the analytical solution are introduced.

Distributions of cracks and contact forces inside the coal model,stimulated by the three input schemes, are exhibited in Fig. 7 (reddots denote tensile cracks, blue dots denote shear cracks and blacklines denote contact forces between particles). It can be seen thatthe contact forces are more concentrated (denoted by the thickerlines) around the fracturing space under stimulation of the pressure

Fig. 5. Three input schemes of

ripples. The coal model stimulated by a constant water pressure(scheme I) only developed a single dominating fracture, withbranch fractures being nearly negligible (Fig. 7a). However, simu-lation results in scheme II and III indicate that both main fracturesand branch fractures are fully developed (Fig. 7b and c). Thesectional views of particle clusters (identified by diverse colours)are exhibited at the lower-right corner, indicating similar phe-nomena observed from crack propagations. From scheme I to III,discrete particles and clusters are increased. The original cleatsintersect with each other and more clusters are produced underpressure ripples.

Significantly, a new type of fracture i.e., ‘spacing fracture’ isgenerated by the pulse loading. Fig. 7d shows the plan view offractures stimulated by scheme III. The spacing fractures are pro-duced around themain fracture (resemble the branch fractures) butare not attached to it. The formation of spacing fractures can be

applied hydraulic pressure.

Fig. 6. Fluctuation of particle velocities stimulated by scheme II (above) and III (below).

Fig. 7. Distributions of cracks and contact forces inside coal model.

C. Ma et al. / Journal of Natural Gas Science and Engineering 46 (2017) 651e663658

explained in two aspects: bond breaking between particles and theintersection of smooth joints (e.g., the No. 1 and No. 2 intersectionsin Fig. 7d), which indicates that the generation of this type offracture is highly affected by the original coal cleat system. As thespacing fractures and branch fractures may gradually mergetogether with the accumulation of cracks, spacing fractures can alsobe viewed as a developmental stage for branch fractures. Obviously,this type of fracture improves the extension of branch fractures sothat a better fracture network can be formed from the PHF effect.Field PHF test exhibits similar fracture networks with the

simulation results (Fig. 8). The fully developed spacing fracturesand branch fractures extend the fracture network, which greatlyincreases the permeability of coal reservoirs.

The spacing fracture is a unique fracture type from the PHFstimulation. It is not directly resulted from the increase of waterpressure but rather generated by the energy of pressure ripples thatare conducted from the fracturing fluid into the surrounding coalmass. Pressure ripples continuously pound the particles and formserials of excitation so that the wave energy can be transported intothe coal mass. During the constant stimulation, cracks are gradually

Fig. 8. Expected effect of pulsating hydraulic fracturing in field test (Li et al., 2015): (a) crack extension at first (b) fracture network at last.

C. Ma et al. / Journal of Natural Gas Science and Engineering 46 (2017) 651e663 659

developing into spacing fractures. A similar phenomenon cannot beobserved from scheme I.

3.3.2. (2) Development of hydraulic pressureDevelopment of hydraulic pressure (in scheme III) approaching

the fracturing space is recorded in Fig. 9 and two unbroken hostingdomains for hydraulic pressures (H1& H2 in Fig. 4b) are selected toillustrate. Firstly, hydraulic pressures are increasing in vibration andthe vibration becomes more intensive when hydraulic pressuregrows larger. This is caused by a hydraulic-mechanical couplingeffect of the second term in equation (12), which represents thatthe mechanical change in domain volume causes a change indomain pressure, stimulated by the periodic loading of pressureripples. The peak pressure in vibration necessarily has a majorimpact on crack behaviours.

Considering the PHF effects and the overall dynamic stability ofthe coal model, the seepage range is setting to be confined to thefracturing space (refer to the last paragraph in 3.2). The strength-ened crack behaviours caused by pressure vibration in domains arebelieved to exist close to the fracturing space. Moreover, the com-bined dynamic actions of the pressure ripples (from the fracturingspace) and the pressure vibrations (from the flow process) aremuch complicated and need more consideration in future works.

3.3.3. (3) Fracturing process and energy evolutionStrain energy stored between particles will be released when

cracking occurs. The indicator of “crack intensity” is proposed tomeasure the energy release so that fracturing effects can be prop-erly discussed. In laboratory experiments, this type of energy isdetected by using an acoustic energy (AE) monitor that is attachedto the surface of the rock sample. In this paper, crack intensities(assumed to be the complete release of the peak strain energy) aredirectly measured and located inside the coal model, which can becalculated based on:

Fig. 9. Measurements of hydraulic pressures inside coal model.

Es ¼ f 2n2kn

þ f 2s2ks

; (14)

where Es denotes the strain energy; fn and fs denote the normal andshear contact forces, respectively; kn and ks denote the normal andshear stiffness, respectively.

Besides, advanced indicators are proposed to quantify the frac-turing effects in simulation. Crack events density (CED) denotes theevent counts at each calculation step and has a proportional rela-tionship with crack intensity rate (CIR). CIR denotes the result ofaverage crack intensity plus event counts at each calculation step.Kinetic energy (KE) is measured as the summation of all particles’kinetic energy in the coal model:

Ek ¼ 12

XNb

miV2i þ Ii

�u2i

�; (15)

where Nb denotes a number of particles, mi denotes the inertialmass, Ii denotes the inertial vector, Vi denotes the average velocityandui denotes the rotational velocity. Because of the inclusion of allparticles, KE can accurately detect the fluctuation and accumulationof energy for the whole model. These three indicators CED, CIR andKE can effectively recognize the characteristics of crack events overthe time history (or the progress of calculation steps).

Figs. 10a, 11a and 12a exhibit the evolutions of indicators by thethree pressure schemes. In scheme I, the crack events and KE onlyappear at the early stage, which indicates that the coal model entersinto a stable state after a short fracturing duration. However, in PHFschemes II and III, CED and CIR constantly develop and presentrelatively larger values throughout thewhole fracturing process. KEevolutions have a continuous fluctuating pattern due to pressureripples and can be clearly distinguished and divided into fourstages:① initial disturbing stage② continuous developing stage③unstable stage ④ post-unstable stage. The initial disturbing stagerepresents the early fracturing, during which kinetic energy accu-mulates and shows a drastically changing pattern. In the continuousdeveloping stage, cracks are constantly generated inside the coalmodel and KE presents a stably changing pattern. When the wholecoal model ruptures after the accumulation of cracks, this is goingthrough the unstable stage; hence, the changing patterns of CED,CIR, and KE become unstable. The post-unstable stage characterizesthe behaviour of the coal model after the rupture with distinctivelylower KE, indicating that the coal model loses the capability toreceive energy frompulse loading. Previous laboratory experimentspresent similar energy evolution with the simulation results (Liet al., 2013; Zhai et al., 2015). According to the results from Li et al.(2013), the AE energy rate monitored in the sample exhibits as thesame developmental stages as the KE in the simulation.

Fig. 10. Fracturing process and energy evolution in scheme I.

Fig. 11. Fracturing process and energy evolution in scheme II.

Fig. 12. Fracturing process and energy evolution in scheme III.

C. Ma et al. / Journal of Natural Gas Science and Engineering 46 (2017) 651e663660

Figs. 10b, 11b and 12b exhibit the spatial distributions of crackintensity by the three schemes. The maximum energy intensity ofthe three schemes is quite close. Compared with the other two,

scheme III developed a higher scattered distribution and the energycan be detected relatively far away from the center of the coalmodel. Although the distributions of crack events in scheme II and

Fig. 13. Distributions of cracks and contact forces under different frequencies.

C. Ma et al. / Journal of Natural Gas Science and Engineering 46 (2017) 651e663 661

scheme III are similar over the time history, the fracturing effectsinside the coal models are quite different. CED and CIR in scheme IIIare higher than in scheme II, which indicates the stronger frac-turing effects of scheme III. The higher KE can be observed inscheme III than in scheme II for all stages. The unstable stage inscheme III occurs a bit earlier than in scheme II. All of these phe-nomena are resulting from the high frequency components ofpressure ripples and stronger cracking is produced duringsimulation.

Fig. 14. Fracturing process and deve

4. Parametric effects of PHF and discussion

According to the above sections, the analytical solution forpressure ripples is obtained and the fracturing mechanism of PHF isdiscussed for different pulsating schemes. The simulation resultsindicate that pressure ripples stimulated by PHF have a uniquefracturing effect on the coal mass. Particularly, the high frequencycomponents, which are crucial but ruled out from the commonunderstanding of pressure propagation, have a strong impact onthe fracturing process and mass particles’ behaviours. In this sec-tion, fracturing effects affected by different PHF parameters arestudied based on the scheme III. Two important parameters i.e.,frequency and amplitude, which strongly affect the fracturingprocess, are tested and discussed. These parameters are also rela-tively convenient to be controlled in actual engineeringapplications.

4.1. Fracturing effects by different frequencies

This study considers three different PHF frequencies (20 Hz,25 Hz, and 30 Hz, which are equivalent to 56, 70 and 84 cycles ofpressure ripples, respectively) with the same ripple amplitude.Fig.13 indicates that under the same average pressure (20MPa) andripple amplitude (2 MPa) (based on the input function f(t)), higherfrequency of PHF can result in a larger expanded fracture network(fracturing result for the case of 20 Hz can be found in Fig. 7c);

lopments of CED, CIR, and KE.

Fig. 15. Distributions of cracks and contact force under different amplitudes.

C. Ma et al. / Journal of Natural Gas Science and Engineering 46 (2017) 651e663662

particularly, more spacing fractures can be developed.It can also be observed from Fig. 12a and Fig. 14 that crack events

appear more frequently over time history with the increase of pulsefrequency; however, the maximum values of CED and CRI are veryclose in each case. The values of KE indicate that the increase ofpulse frequency mainly affect the fracturing progress in thecontinuous developing stage, during which a higher frequency re-sults in a higher vibrating pattern of kinetic energy.

Although more cracks are developed under higher pulse fre-quencies, the whole progress is not accelerated and the unstable

Fig. 16. Fracturing process and deve

and post-unstable stages accrue approximately at the same time.The simulations conclude that relatively higher frequency canproduce a better fracturing effect in actual applications but may notshorten the fracturing duration. The results do not necessarilyindicate that very high frequency should be used because the en-gineering efficiency needs to be taken into consideration.

4.2. Fracturing effects by different amplitudes

The amplitude of pressure ripples, which can be directlycontrolled by the piston pump of the PHF device, is also a crucialfactor for fracturing effects. Similarly, the study (based on schemeIII) numerically simulates fracturing effects under three differentamplitudes (DPa ¼ 2 MPa, 3 MPa and 4 MPa) with the same fre-quency. Fig. 15 indicates the fracturing results with the increase ofamplitude: the fractured areas become more concentrated, thenumber of spacing fractures are largely decreased and the distri-bution of cracks are not obviously expanded (fracturing results forthe case of DPa ¼ 2 MPa can be found in Fig. 7c).

Moreover, Figs. 12a and 16 indicate that with the increase ofripple amplitudes, the magnitudes of crack events (CED and CIR)are obviously increasing; however, crack events appear lessfrequently compared with the records in Fig. 14 throughout thetime history. The increase of amplitude also affects the continuousdeveloping stage, during which KE is increasing with higheramplitude. More importantly, KE evolutions show that the unstable

lopments of CED, CIR, and KE.

C. Ma et al. / Journal of Natural Gas Science and Engineering 46 (2017) 651e663 663

and post-unstable stages appear earlier in time history, which in-dicates that relatively higher amplitude may accelerate therupturing progress of the coal model and make the fracturingduration shorter.

Note that higher ripple amplitudes can effectively accelerate therupturing progress and shorten the fracturing time. However, abetter fracture network does not necessarily to be produced andspacing/branch fractures are not necessarily developed).

5. Conclusions

In this study, an advanced mathematical model for pressureripple propagation is proposed and an analytical solution is ob-tained. Unlike traditional understanding, which believes that pul-sating hydraulic fracturing (PHF) creates simple sinusoidal pressureripples inside a fracturing pipe, an actual pressure changing patternis more drastic with high-frequency components. In laboratoryexperiments, the high-frequency components are naturally pro-duced because of the strong wave impedance materials; this waveform can be generated by using ‘seal-release’ pulsating scheme.Eventually, this solution is applied for the pulsating loadings innumerical simulations.

Based on the discrete element method, hydraulic fracturing (HF)and PHF fracturing effects are numerically simulated in compara-tive situations. The simulation results indicate that compared withthe traditional HF loading scheme, the schemes with pressureripples can create a better distribution of cracks and fractures.These unique effects can be amplified by introducing the high-frequency components of ripples. The pulsating energy propa-gates through a fracturing pipe and enters into coal mass to pro-duce a new type of fracture (i.e., spacing fracture). This type offracture can intersect with branch fractures and original cleats ofthe coal mass and result in a well-distributed fracture network.Advanced indicators that include crack event density (CED), crackintensity rate (CIR) and kinetic energy (KE) are proposed andmeasured to clarify the conclusions of PHF effects. Particularly, theKE evolution helps to divide the development of fractures underPHF into four stages (① initial disturbing stage ② continuousdeveloping stage ③ unstable stage ④ post-unstable stage).

Furthermore, fracturing effects with different parameters (i.e.,pulsating frequency and ripple amplitude) are simulated and dis-cussed for engineering optimization. Simulation results suggestthat relatively higher pulsating frequency can stimulate a better-distributed fracture network; however, the fracturing durationmay not be significantly shortened by this strategy. Higher rippleamplitude can accelerate the fracturing progress and rupture of thecoal model (unstable and post-unstable stages) appears earlierwith higher amplitudes. However, this phenomenon may havenegative effects on the distribution and extension of a fracturenetwork.

Funding

This research was supported by National Natural ScienceFoundation of China (Grant No. 41230635) and the open fund(Grant No. SKLGP2015Z004) from State Key Laboratory of Geo-hazard Prevention and Geo-environment Protection, China.

Acknowledgements

The authors would like to thank all of the members in ‘Multi-scale, Multiphysics Modelling’ (M3) group of the Centre for Geo-science Computing, University of Queensland; this research couldnot be completed without their valuable assistance andencouragement.

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