Journal of Petroleum Science and Engineering 111 (2013) 71–77

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Journal of Petroleum Science and Engineering

0920-41http://d

n CorrE-m

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

Stimulation of the natural fracture system by gradedproppant injection

Aditya Khanna a, Alireza Keshavarz b, Kate Mobbs b, Michael Davis a, Pavel Bedrikovetsky b,n

a School of Mechanical Engineering, The University of Adelaide, SA 5005, Australiab Australian School of Petroleum, The University of Adelaide, SA 5005, Australia

a r t i c l e i n f o

Article history:Received 22 May 2012Accepted 15 July 2013Available online 24 July 2013

Keywords:Coal bed methaneNaturally fractured reservoirsWell stimulationGraded proppant injectionInjection scheduleMathematical modelling

05/$ - see front matter & 2013 Elsevier B.V. Ax.doi.org/10.1016/j.petrol.2013.07.004

esponding author. Tel.: +61 8 8313 3082.ail address: pavel@asp.adelaide.edu.au (P. Bed

a b s t r a c t

Graded proppant injection i.e. the injection of particles of increasing size and decreasing concentration, isproposed as a method for stimulating natural fractures. Compared to the injection of mono-sizedproppant, graded proppant injection would facilitate deeper percolation of proppant into the fracturesystem and enhanced well productivity.

A mathematical model is developed which describes the injection stage and the capture kinetics ofproppant particles in the natural fracture system. Based on the mathematical model, an injectionschedule is developed which would result in the optimal placement of proppant in the fracture system. Acase study is conducted to estimate the change in well productivity index due to the application of thegraded particle injection method.

& 2013 Elsevier B.V. All rights reserved.

1. Introduction

Naturally fractured reservoirs such as coal bed methane (CBM),tight and shale gas fields and geothermal reservoirs often requiresome form of stimulation in order to achieve economical produc-tion rates. Fluid injection at the wellbore prior to production is awidely adopted technology for stimulating natural fractures(Colmenares and Zoback, 2007). The injection of fluid withoutproppant may lead to an increase in permeability due to the shearinduced opening of rough natural fractures (Hossain et al., 2002;Rahman et al., 2002). Injection of proppant on the other hand,leads to an increase in reservoir permeability by preventing theclosure of the dilated natural fractures (Holditch et al., 1968;Warpinski et al., 2008). However, in the absence of a widehydraulic fracture, the large proppant particles typically used infracturing treatments can only stimulate a small region in thevicinity of the wellbore. This is due to the rapid decline in theaperture of stimulated natural fractures with increasing distancefrom the wellbore.

In this paper, the method of graded proppant injection isproposed. Injecting small particles first followed by larger particlescould increase the size of the stimulated zone, since the smallparticles are more likely to percolate deeper into the fracturesystem. The main parameters of interest in the problem of gradedproppant injection are: the increase in well productivity and the

ll rights reserved.

rikovetsky).

proppant injection schedule i.e. the dependence of particle sizeand concentration on the injection time. Within this paper, asimplified engineering model is developed for calculating theseparameters.

The problem of proppant placement in a fracture system isformulated in Section 2. A mathematical model is then presentedwhich describes: the opening of natural fractures due to fluidinjection (Section 3.2), particle plugging in the opened fractures(Section 3.3), the optimal proppant placement in the fracturenetwork (Section 3.4) and the injection schedule (Section 3.5).The change in well productivity is calculated in Section 3.6. A casestudy is conducted in Section 4 using typical reservoir data todetermine the effect of injection rate, and size of stimulated zoneon the change in well productivity index.

2. Problem formulation

The proppant placement resulting from graded proppant injec-tion is schematically presented in Fig. 1. It shows the straining ofparticles of different sizes in the fracture network. The basicfeatures of the injection schedule which yields a dependence ofthe injected particle size and concentration upon the injectiontime, can be deduced from Fig. 1. During fluid injection at thewellbore, the fluid pressure and consequently, the opening offractures diminishes with increasing distance from the well. Thisimplies that smaller particles need to be injected before largerparticles in order to achieve deeper percolation. In terms of theinjection schedule, this implies that the injected particle radius

Fig. 1. Plugging of the natural fracture system by graded proppant particles.

Deformed flow channel

Rigid proppant Fluid

Normal stress

Normal stress

Fig. 2. Two competitive factors affect the hydraulic resistance: (1) The deformationof the flow channel caused by rock stresses and (2) the additional tortuosity of theflow path due to the presence of the particles.

A. Khanna et al. / Journal of Petroleum Science and Engineering 111 (2013) 71–7772

must increase with injection time. Also, a greater number of cleatsare encountered by the flow with increasing distance from thewellbore. Thus, a greater concentration of small particles is neededto plug the filtration path compared to large particles. Conse-quently, the concentration of injected particles must decrease withinjection time.

There are two competitive mechanisms, shown in Fig. 2, whichaffect the permeability of the stimulated fracture system. Theseare: the reduction in the fracture aperture due to the confiningstresses and the additional tortuosity of the flow path due to thepresence of proppant particles. A dense packing of the proppantparticles minimizes the deformation of the fracture and a sparsepacking minimizes the additional tortuosity to the flow path. Anintermediate proppant concentration exists, at which net effect ofthese two competitive mechanisms is minimum. At this optimumproppant concentration, the maximum conductivity of the fracturesystem is achieved.

The optimal proppant concentration depends upon a numberof parameter such as the reservoir stress, material properties ofthe rock and the strength of the proppant. In order to determinethe value of the optimal proppant concentration and the fractureconductivity during production, experiments similar to Brannonet al. (2004) can be conducted for specific rock and proppant

combinations. For simplicity, the semi-analytical results obtainedby Khanna et al. (2012) are utilized here. In the next section, themathematical model for particle straining in fractured porousmedia is presented.

3. Mathematical model for flow in fractured porous mediawith strained particles

3.1. Modelling assumptions

The following assumptions are utilized regarding graded prop-pant injection:

�

At the scale of the reservoir, the flow from the wellboretowards the fracture system is axi-symmetric and can bedescribed by Darcy's law for radial flow. The conductivefractures form a tree-like or dendritic structure with branchesmultiplying from the well towards the reservoir.

�

During injection, the flow is steady state since the time ofpropagation of an elastic pressure wave in the stimulated zoneis negligibly small compared to the total time of injection. Thecalculations are performed for constant injection rate at thewellbore.

�

The naturally fractured reservoir is idealized by the matchstickmodel in which match-like blocks of rock matrix are separatedby cleats or fractures. The height of the matrix blocks is equal tothe height of the reservoir and the width of the blocks is equalto the average cleat spacing.

�

The dependence of the permeability of the cleat system uponpressure is obtained from a widely used permeability modelsuggested by Palmer and Mansoori (1998). The terms withinthe Palmer–Mansoori model related to matrix swelling/shrink-age are neglected since the injection period is relatively short.

�

In order to simplify the particle capture criterion, the presenceof fracture wall asperities, in-fill material and tortuosity in thefractures are neglected. It is assumed that particle strainingoccurs when the particle diameter exceeds the pressure-dependent opening of the fracture.

�

The particles are assumed to travel at the same speed as theinjection fluid, i.e. proppant settlement by gravity is neglected.

The mathematical model for water injection before particleplugging, based on the above assumptions, is presented in the nextsection.

3.2. Water injection stage

In this section, the basic equations for water injection intonaturally fractured system are presented. Darcy′s law for radialflow of an incompressible fluid through deformable rock is givenby

q2πr

¼� kðpÞμ

dpdr

ð1Þ

Here q is the constant injection rate per unit thickness of thereservoir, pðrÞ is the pressure at a distance r from the wellbore, kðpÞis the pressure dependent permeability of the fracture networkand μ is the fluid viscosity. Assuming a given pressure dependencefor permeability and separating variables in the ordinary differ-ential equation (1) yields the implicit formula for pressure dis-tribution around the well:

qμ2π

lnrer¼Z p

pres

kðpÞdp; ð2Þ

A. Khanna et al. / Journal of Petroleum Science and Engineering 111 (2013) 71–77 73

Here re is the drainage radius at which the fluid pressure is equalto the initial reservoir pressure pres. The pressure–permeabilityrelationship substituted into Eq. (2) is

kðpÞ ¼ ko 1þ Cϕo

ðp�presÞ� �3

ð3Þ

This expression comes from the model for permeability proposedby Palmer and Mansoori (1998). The constants ko and ϕo are theinitial permeability and porosity of the reservoir, respectively andthe constant C represents the reservoir compressibility underuniaxial strain. It can be expressed in terms of the Young′smodulus E and Poisson′s ratio ν of the reservoir as

C ¼ ð1þ νÞð1�2νÞð1�νÞ

1E

ð4Þ

Substituting Eq. (3) into (2) and integrating in p yields the explicitformula for the radial pressure distribution during water injection:

pðrÞ ¼ pres þϕo

C2qμCπkoϕo

lnrerþ 1

� �1=4

�1

" #; pðrwÞ ¼ pw ð5Þ

The pressure at the wellbore radius rw is denoted by pw.The radial coordinate r can be normalized against the drainage

radius of the reservoir to yield the dimensionless radial coordinaterD ¼ r=re and the dimensionless wellbore radius rwD ¼ rw=re.Further, a dimensionless injection rate εq can be defined as

εq ¼ 2qμCπkoϕo

ð6Þ

In terms of these dimensionless parameters, the expression for theradial pressure distribution can be re-written as

pðrDÞ ¼ pres þϕo

C1þ εqln

1rD

� �1=4

�1

" #; pðrwDÞ ¼ pw ð7Þ

By substituting Eq. (7) into (3), the permeability distributionaround the well kðrDÞ can be obtained as

kðrDÞ ¼ ko 1þ εq ln1rD

� �3=4

ð8Þ

From the dimensionless analysis of the relationship between therock permeability, mean aperture of fractures and matrix blocksize (Bear, 1972; Basniev et al., 1988) it follows that

kko

¼ hho

� �3

ð9Þ

where ko and ho are the initial permeability of the reservoir andthe initial aperture of the fractures, respectively. Substituting Eq.(8) into relationship (9) results in an expression for the effectivecleat opening given by

hDðrDÞ ¼hðrDÞho

¼ 1þ εq ln1rD

� �1=4

ð10Þ

Here the apertures h and ho are averaged openings incorporatingthe effect of surface asperities. From Eq. (10), it can be observedthat a higher dimensionless rate εq results in a larger fractureopening at any given distance.

Note that the initial fracture opening ho and the fracturesspacing L can be related to the initial permeability and porosityof the fracture system by assuming a regular fracture arrangementin the reservoir. For a matchstick arrangement of matrix blocks,the formulae for permeability and porosity are given by (van Golf-Racht, 1982)

ko ¼h3o12L

; ϕo ¼2ho

L; ð11Þ

3.3. Size exclusion of particles during the injection

In this section, a highly simplified model for proppant particlestraining in the fractures is presented. It is assumed that particlestraining occurs when the cleat aperture hðrÞ is twice the particleradius rs i.e. at the plugging moment:

hðrÞ ¼ 2rs ð12Þwhere the averaged cleat opening hðrÞ is given by Eq. (10). The sizeexclusion condition (12) neglects the asperities on the fracturewalls as well as the tortuosity of the flow path in a network offractures. An alternative size exclusion condition could beγhðrÞ ¼ 2rs, where γo1 is a shape factor. For hydraulic fractures,Valko and Economides (1995) suggest γ ¼ 1=3. For network ofrough natural fractures, the shape factor can be determined eitherexperimentally or by using computational fluid dynamics. How-ever, determining such a shape factor is out of scope of the presentwork and instead, the simple criteria given by Eq. (12) is used.Once calculated, the shape factor can be readily incorporated inthe present formulation.

The size exclusion condition (12) can be used to calculate theplugging distance or the distance travelled by a particle of radius rsbefore being captured. The dimensionless plugging distance fromthe wellbore is given by

rDðrDsÞ ¼ exp1�ð2rDsÞ4

εq

!; ð13Þ

where rDs ¼ rs=ho is the dimensionless particle size. The minimumproppant radius ro is simply obtained by substituting the radius ofthe stimulation zone rst into Eq. (13) and re-arranging to yield:

ro ¼minðrDsÞ ¼ 12

1þ εqln1α

� �1=4

ð14Þ

where α¼ rst=re is the scaled radius of the stimulation zone.The size exclusion condition can also be used to calculate the

travelling time of a particle before capture. The interstitial velocityU of the fluid in a cleat at a distance r from the well is simply givenby

U ¼ qnðrÞhðrÞ ð15Þ

where q is the injection rate per unit thickness of the reservoir, nðrÞis the number of cleats through which the fluid flows and hðrÞ isthe aperture of the cleats. The number of cleats nðrÞ encounteredby the injected fluid increases with increasing distance from thewellbore. To obtain a relationship between the number of cleatsand the distance from the wellbore, consider an idealized fracturesystem comprising of two perpendicular sets of evenly spacedfractures, also known as the matchstick model. A circle of diameter2r >> L would be intersected by a set of evenly spaced fracturesroughly 4r/L times. Hence, for two perpendicular sets of fractures,the number of cleats crossing a circle with radius r is approxi-mated by

nðrÞ ¼ 8rL

ð16Þ

where L is the cleat spacing. Substituting (10) and (16) into (15)and separating the variables results in an expression for theparticle trajectory,

qL8ho

t ¼Z r

rwρ 1þ εqln

reρ

� �1=4

dρ ð17Þ

where t is the time taken by a particle to travel a distance r and ρ isthe dummy integration variable. The above expression assumesthat the particle velocity is the same as the velocity of the carrierfluid and no proppant settlement occurs. In terms of the

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

CFD Results

Best fit

Packing ratio ρ

Cor

rect

ion

fact

or f(

β,ε σ

)

0

10-4

10-3

10-2

Increasing

dimensionless confining stress, εσ

Fig. 4. Correction factor vs. packing ratio for various values of the dimensionlessconfining stress, values indicated above individual curves (Khanna et al., 2012).

A. Khanna et al. / Journal of Petroleum Science and Engineering 111 (2013) 71–7774

dimensionless radial coordinate rD ¼ r=re, the travelling time isgiven by

qL8ho

t ¼ r2e

Z rD

rwD

℘ 1þ εqln1℘

� �1=4

d℘ ð18Þ

Substituting the expression for plugging distance given by (13)into Eq. (18) yields the settling time ts or the time it takes for theparticle of size rDs to strain in the thinning cleat,

tsðrDsÞ ¼8hor2eqL

Z rDðrDsÞ

rwD

℘ 1þ εqln1℘

� �1=4

d℘ ð19Þ

Eq. (19) can be used to calculate the total injection time i.e. thetravelling time for the smallest particle tsðroÞ. It can be observedfrom Eq. (19) that a higher injection rate results in a shortersettling time i.e. faster arrival of a particle to its plugging site. Thetotal injection time also decreases with increasing injection rate.

3.4. Optimal placement of proppant in the cleat system

Let there be some averaged distance lðrsÞ between particles forwhich a parameter β can be defined as

β¼ 2rslðrsÞ

ð20Þ

The parameter β is referred here as the packing ratio. A packingratio equal to zero implies the absence of proppant particles in thecleat system and a packing ratio equal to unity implies that theproppant particles form a full monolayer in the cleat system. Thus,β represents the areal concentration of proppant in a partialmonolayer.

The separation of particles within the monolayer directlyinfluences the permeability of the stimulated fracture. As dis-cussed in Section 2 and shown in Fig. 2, there exists an optimalproppant concentration for which the fracture permeability ismaximized. The optimal value of the parameter β is denoted by βn.For simplicity, we adopt the semi-analytical approach developedby Khanna et al. (2012) to calculate the optimal proppant con-centration. However, it can also be determined by conductingexperiments similar to Brannon et al. (2004) for specific rock andproppant combinations.

The resultant permeability of the fractures within the stimula-tion zone is greater than the initial permeability ko, but less thanthe permeability kðrDÞ of the stimulated fractures before particlestraining. The fracture system permeability inside the stimulationzone can be written as f kðrDÞ where the factor f represents acorrection factor. In the semi-analytical model of Khanna et al.(2012) which is also used here, the factor f was determined as afunction of the packing ratio β and the dimensionless stress in thereservoir εs ¼ sð1�v2Þ=E. The latter parameter incorporates

Flow

Symmetry element

Inlet

M

l

l

Fig. 3. Semi-analytical approach to determine the optimal proppant concentration. (a) Adeformed fracture geometry is calculated using Hertz contact theory and the fracture c

confining stress and the elastic properties of the rock. ClassicalHertz contact theory and computational fluid dynamics were usedto compute the function f ðβ; εsÞ. Fig. 3 shows the model setup andFig. 4 shows the results. The optimal proppant concentration βn,for which the factor f is maximum, can be observed in Fig. 4. It canalso be observed from Fig. 4 that the value of the optimal packingratio βn increases with increasing value of dimensionless stress εs.This is because, a greater concentration of proppant particles isrequired to prevent the closure of the fracture at higher confiningstress.

In terms of the correction factor, the permeability of thefracture system during production is given by

kprðrDÞ ¼f ðβn; εsÞkðrDÞ rD≤αko rD4α

(ð21Þ

where kðrDÞ is permeability of the fractures before plugging, givenby (8) and α¼ rst=re is the scaled stimulation radius. For knownreservoir stress and elastic properties, the optimal value of packingratio βn and the corresponding value of the factor f ðβn; εsÞ can beobtained from Fig. 4. During fluid injection, the hydraulic pressurekeeps the fractures open i.e. f is a function of βn only. For theoptimal value of packing ratio βn, the value of f ðβn;0Þ can be readfrom the εs ¼ 0 curve in Fig. 4. Thus, the permeability of the

Outlet

Flow streamlines

easurement planes

regular arrangement of proppants between the fracture faces is considered. (b) Theonductivity is calculated by performing CFD analysis (Khanna et al., 2012).

0

20

40

60

80

100

0 100 200 300 400 500

Cle

at a

pertu

re h

(r) (

µm)

Radial distance from the wellbore r (m)

Increasing injection rate

ho

Fig. 5. The distribution of cleat opening as a function of the radial distance fromthe wellbore.

0

4

8

12

16

20

24

0 100 200 300 400 500

Settl

ing

time

t s(r)

(hou

rs)

Radial distance from the wellbore r (m)

Increasing injection rate

Fig. 6. Particle settling time as a function of the radial distance from the wellbore.

A. Khanna et al. / Journal of Petroleum Science and Engineering 111 (2013) 71–77 75

fracture system during injection is given by

kinðrDÞ ¼f ðβn;0ÞkðrDÞ rD≤αkðrDÞ rD4α

(ð22Þ

3.5. Calculation of the optimal injection schedule

If the total injection time is taken as the settling time of thesmallest particle, the injection time is given by

tinðrDsÞ ¼ tsðroÞ�tsðrDsÞ ð23ÞSubstituting (23) into (19), the time at which a particle of size rDsshould be injected can be obtained as

tinðrDsÞ ¼8hor2eqL

Z α

rDðrDsÞ℘ 1þ εqln

1℘

� �1=4

d℘ ð24Þ

The schedule for the injected particle radius vs. time rDsðtinÞ isobtained by inversion of the above function.

The required number of particles of a given size, Np can beestimated as

NpðrDsÞ ¼H

lnðrDsÞnðrÞ ¼ βn H

L4r

rDshoð25Þ

where βn is the optimal packing ratio, H is the height of thereservoir and L is the spacing between the cleats. Using particleradius rDs as a parameter in Eqs. (25) and (24), the schedule forinjected particle concentration vs. time, NpðtinÞ can be obtained.The optimal injection schedule rDsðtinÞ and NpðtinÞ depends on thedimensionless injection rate εq, the size of the stimulation zonerst=re and the value of the optimal packing ratio βn. The value ofoptimal packing aspect ratio together with the effective perme-ability of the proppant monolayer allows determining well indexduring injection and production.

3.6. Calculation of injectivity and productivity index with pluggedcleats

The formula for the well injectivity index before particlestraining is obtained by substituting Eq. (8) into (1) followed.Separating the variables and integrating both sides yields:

Io ¼ qpw�pres

¼ 2πkoμ

Z 1

rwD

drDrDð1þ εqlnð1=rDÞÞ3=4

" #�1

¼ πko2μ

εqhDðrwDÞ�1

ð26ÞIn the above expression, hDðrwDÞ is the normalized opening at thewellbore. Similarly, the well injectivity after particles straining canbe calculated by substituting (22) into (1).

I ¼ πko2μ

f ðβn;0ÞεqðhDðrwDÞ�hDðαÞÞ þ f ðβn;0ÞðhDðαÞ�1Þ ð27Þ

When the entire flow region is stimulated, the fracture opening atthe stimulation radius is equal to the initial fracture aperture, i.e.hDðαÞ ¼ 1. In this case, the injectivity index after plugging is simplygiven by I ¼ f ðβn;0ÞIo.

The productivity index during liquid (water) production can becalculated by substituting (21) into (1):

J ¼ 2πkoμ

f ðβn; εqÞεq4ðhDðrwDÞ�hDðαÞÞ þ f ðβn; εqÞεq lnð1=αÞ

ð28Þ

When the entire flow region is stimulated, the productivity indexafter plugging is simply given by J ¼ f ðβn; εqÞIo. The productivityindex can be normalized against the productivity index of anunstimulated reservoir:

Jo ¼2πkoμ

1lnð1=rwDÞ

ð29Þ

The normalized productivity index is then given by JD ¼ J=Jo. Thenormalized productivity index represents the ‘folds of increase’ inproduction due to graded proppant injection.

4. Case study and numerical results

In order to quantify the requirements and benefits of theproposed stimulation method, a case study is conducted on ahypothetical coal seam gas reservoir using typical reservoir para-meters provided by Moore (2012) and Laubach et al. (1998).

At a dimensionless confining stress εs ¼ 10�3, the value of theoptimal packing ratio can be read from Fig. 4 as βn ¼ 0:24. Thecorresponding values of the correction factors are also read fromFig. 4 as f ðβn;0Þ ¼ 0:78 and f ðβn; εqÞ ¼ 0:55. The calculations areperformed for three values of the dimensionless injection rate. Thechosen values of injection rates per unit thickness are 1, 5 and10 bbl/min (where 1 bbl/min¼2.65E�3 m3/s). The correspondingvalues of the dimensionless injection rate εq are approximately 2,10 and 20. The dimensionless injection rate was calculated usingEq. (6).

For the three different values of the dimensionless injectionrate, the radial distribution of the cleat opening is calculated usingEq. (10). The results are shown in Fig. 5. The change in fracture

0

10

20

30

40

50

0 5 10 15 20 25

Parti

cle

radi

us r s

(µm

)

Injection time tin (minutes)

Increasing injection rate

0

2

4

6

8

10

12

14

0 5 10 15 20 25

No.

of p

artic

les N

p(r s

) x10

7

Injection time tin (minutes)

Increasing injection rate

Fig. 7. Proppant injection schedule for stimulating a 50 m radius around the wellbore: (a) particle radius vs. time and (b) particle concentration vs. time.

1

3

5

7

9

0 100 200 300 400 500

Nor

mal

ized

pro

duct

ivity

inde

x JD

Stimulation radius rst (m)

Increasing injection rate

Fig. 8. Normalized productivity index or folds of increase in production as afunction of stimulation radius and injection rate.

A. Khanna et al. / Journal of Petroleum Science and Engineering 111 (2013) 71–7776

aperture is very rapid near the wellbore at any given injection rate.The fracture opening increases with increasing injection rate.

Next, the travelling time is calculated as a function of the radialdistance using Eq. (18). The results are shown in Fig. 6. The slope ofthe curves increases with distance from the wellbore whichimplies that the interstitial velocity of the fluid in the cleatsdecreases with increasing distance from the wellbore. It is due tothe branching structure of the fracture system. A higher injectionrate results in the quicker arrival of particles at the plugging site. Inthe present example, a particle would take approximately 21 h toreach the drainage radius of 500 m at an injection rate of 1 bbl/min, whereas it would take less than 4 h to reach the drainageradius at an injection rate of 10 bbl/min.

The injection schedule is calculated based on a pre-determinedsize of the stimulation zone. In the present example, the radius ofthe stimulation zone is chosen to be 50 m. Hence, the value of thescaled stimulation radius is α¼ rst=re ¼ 50=500¼ 0:1. For α¼ 0:1and an optimal packing ratio βn ¼ 0:24, the injection schedule iscalculated using Eqs. (24) and (25). The results are shown in Fig. 7.Fig. 7(a) shows the dependence of particle radius on injection timeand Fig. 7(b) shows the dependence of particle concentration onthe injection time. As discussed in Section 2, the particle radiusincreases with injection time. Conversely, the particle concentra-tion decreases with injection time. The rapid increase in theparticle size and the rapid decrease in the particle concentrationnear the end of the injection period are due to the change infracture opening and fluid velocity near the wellbore. It can beobserved that the injection rate significantly affects the injectionschedule. With increasing rate, the capture time of the particles isreduced, hence the total time of injection decreases. In the presentexample, stimulating a 50 m radius around the wellbore requiresapproximately 22 min at an injection rate of 1 bbl/min but lessthan 4 min at an injection rate of 10 bbl/min. The required particlesize increases with injection rate since larger fracture openings areproduced. The injected particle concentration decreases withincrease in injection rate since the inter-particle separation mustincrease with increasing particle radius to obtain the optimal valueof packing ratio (see Eq. (20)).

The productivity index is calculated using Eqs. (28) and (29).Fig. 8 shows the dependence of the normalized productivity indexJD, which represents the ‘folds of increase’ in production, oninjection rate and the radius of the stimulation zone rst . A higherinjection rate results in a greater well productivity index for allvalues of the stimulation radius. The injection rate also controlsthe relative increment in productivity for a given increment in the

radius of the stimulation zone. For example, at a low injection rateof 1 bbl/min, the productivity index does not increase significantlybeyond a stimulation radius of 50 m. At a high injection rate of10 bbl/min, the productivity index continues to increase withincreasing size of the stimulation zone. It can be concluded fromFig. 8 that the graded proppant injection scheme is a promisingtechnology for the stimulation of natural fractures. In the exampleconsidered here, a three-fold increase in production was obtainedby stimulating a radius of 50 m around the wellbore at a rate of5 bbl/min. The limitations and applicability of the current math-ematical model are discussed in the next section.

5. Discussion

Some radical simplifications were adopted to model the inter-action of fluid with natural fractures as well as particle transport inthe fracture. Some shortcomings of the present model andpossible methods for improvement are:

�

Axi-symmetric flow geometry was assumed in the mathema-tical model (Eqs. (1) and (15)). The flow geometry simplificationimplies that the conductive fractures form a tree-like

A. Khanna et al. / Journal of Petroleum Science and Engineering 111 (2013) 71–77 77

(dendritic) structure with branches multiplying from the welltowards the reservoir. This assumption is equivalent to parallellayers (tubes) for the plane-parallel flow, which is often usedfor estimating flow in porous media. A more precise descriptionof flow in fractured media would require the application ofpercolation or effective media models.

�

The detailed description of the proppant flow and capture in realfractured system certainly requires more elaborate modelling.Bedrikovetsky (2008) described the size exclusion of suspensionflow in stochastic porous media. To the best of our knowledge, themodel for suspension flow in fractured systems is not available inthe contemporary literature. However, further development of theproposed technology must include modelling of the straining-dominated suspension transport in stochastic fractured systems(Wei et al., 2012).

�

In order to apply the technique of graded particle injection incoal beds, the anisotropy of coal needs to be incorporated in themodel to describe particle propagation in face and butt cleats.The elaborated theory must include the tensorial permeabilityof the cleat system and the capture criterion should also bedirection-dependent.

�

Non-elastic processes of the proppant crushing or embedmentmay result at high stresses if the rock is significantly harder orsofter than the proppant. As a result, a partial proppantmonolayer may provide inadequate fracture conductivity. Toincorporate the effects of these phenomena on the optimalproppant concentration, experimental studies must be con-ducted on specific rock-proppant combinations under expectedstress conditions.

�

The injection of water changes the reservoir stress around thewellbore, often causing shear induced slip of the asperousfractures, which may lead to additional productivity enhance-ment. The effect of shear dilation must be included in themodel for graded particle injection for stimulation of naturalfractured systems.

6. Conclusion

A simple mathematical model is proposed for studying thegraded proppant injection in natural fracture systems, such asthose in coal seam beds, shale gas reservoirs and geothermalfields. The case study conducted using typical reservoir datademonstrates that graded proppant injection may lead to theenhancement in well productivity. It was found that the mostinfluential parameter affecting the proppant injection schedule

and the well productivity index is the fluid injection rate. A higherinjection rate results in greater opening of the cleats, deeperpercolation of the smallest particles, shorter injection time and agreater increase in well productivity. The applicability of thepresent model is limited due to a number of radical simplifica-tions. Better estimates of increase in well productivity can beobtained by incorporating more advanced models for particletransport in fractured media.

Acknowledgements

The authors wish to thank Prof. Sheik S. Rahman from TheUniversity of New South Wales and acknowledge the support ofSantos Ltd. and ARC linkage grant LP110200799.

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