Propagation and aperture of staged hydraulic fractures in unconventional resources in toughness-dominated regimes

Ali Tghihin,Hmid Hshemlhoseini,Mushrrf Zmn,Sied Beheshti Zvreh

aDepartment of Civil Engineering,Isfahan University of Technology,Esfahan,Iran

bMewbourne School of Petroleum and Geological Engineering,Sarkeys Energy Center,University of Oklahoma,Norman,USA

cDepartment of Mining Engineering,Isfahan University of Technology,Esfahan,Iran

1.Introduction

In unconventional completion,single-stage fracture treatments evolved to multistage stimulation treatments and fracturing of standalone wells was progressed to simultaneous fracturing of multilateral wells in order to increase reserves per well,enhance well productivity,and improve project economics(King et al.,2008).Some researchers mentioned that an optimization is required for such completion techniques.For example,it was reported that increase in spacing between the fractures induces less interference betweenpropagating fractures and hence requires lessbreakdown pressureto initiateafracture(Singh and Miskimins,2010).In addition,no deviation or collapse of hydraulic fractures occurs under this circumstance.It is the reason of considering stress shadow concept in designing of hydraulic fracturing pattern for this kind of reservoirs.Stress shadowconcept has been discussed by many researchers(e.g.Roussel and Sharma,2011;Morrill and Miskimins,2012;Taghichian et al.,2014).

The main purpose of hydraulic fracturing in unconventional reservoirs is to create parallel hydraulic fractures perpendicular to the horizontal wellbore and make a connected reservoir to the wellbore.In order to reach this goal,observing whether a hydraulic fracture propagates or is trapped is of paramount importance.In addition,the direction of propagation of hydraulic fractures plays a decisive role in having ideal straight hydraulic fractures with no deviation and collapse.Therefore,a fracturing pattern should be designed in such a way that simultaneously satisf i es both of the above-mentioned conditions,i.e.reasonable fracturing pressure(via stress intensity factor(SIF))and controlling the fracturing direction(via stress shadow effect(SSE)).

Propagation of hydraulic fractures depends on stress field around the tip.Hence,many researchers have tried to calculate the stress field around cracks which is influenced by crack geometry,fracturing scheme,and applied boundary conditions.In order to simplify the stress calculation close to the tip,a new term called SIF was defined.The generalized form of stress field equation for modes I,II and III of loading,valid in the vicinity of the crack tip,is given by

whereσijis the stress in the vicinity of the crack tip;f,gandhare the trigonometric functions;randθare the cylindrical coordinate components;andKI,KIIandKIIIare the modes I,II and III SIFs,respectively.The concept of SIF was first proposed by Irwin(1957),by which the effects of geometry and boundary conditions are separated from spatial location of stress analysis.

Assuming a planar crack in thex-zplane,one can have the SIFs as

One conspicuous point herein is the dependence ofKIIandKIIIon the shear stresses.When crack propagating,the angle of propagation in two dimensions depends onKIandKIIwhich can be obtained via the following relationship(Stone and Babuska,1998):

whereΘis the change angle of propagation direction.It is induced from Eq.(3)that no change in the propagation direction is observed in case of having noKII,which corresponds with the case of having no shear stress at the tip(see Eq.(2b)).Therefore,it can be induced that having straight hydraulic fractures with no deviation or collapse needs negligible shear stresses at the tip.It is noted that this is an ideal condition which is assumed for the current analyses in order to obtain the ideal distance between hydraulic fractures propagating merely under the circumstance of mode I.Therefore,under the above-mentioned condition,the only required term to determine stress field around the crack tip isKI,which changes with geometries of the crack,medium,boundary and loading conditions(Broek,1982).

In this way,having the SIF behavior of hydraulic fractures,one can judge how/when a hydraulic fracture propagates in the medium.Many two-dimensional(2D)crack geometries have been analytically solved and their SIFs have been reported in the literature(e.g.Tada et al.,2000).However,there have also been some problems in which the geometries of cracks/medium were challenging and stress field for these fractures was not easy to be analytically determined.The SIF of such problems was defined by utilizing boundary element(BE)and finite element(FE)methods(e.g.Sih,1973;Murakami,1987;Tada et al.,2000).Threedimensional(3D)cracks with different geometries,such as embedded cracks in an infinitely extended homogeneous,isotropic solid medium,opened up due to prescribed internal pressure,have also been analyticallysolved bya number of investigators(e.g.Keer,1964;Sneddon and Lowengrub,1969;Shah and Kobayashi,1971;Guidera and Lardner,1975).Mastrojannis et al.(1979)also developed a method for determination of SIF for a general-shaped crack with internal pressure in an infinite medium utilizing numerical integration.Furthermore,using the 2D Fourier transform method,Kassir(1981)succeeded in solving the SIFs around rectangular cracks.Nejati et al.(2015)also proposed a novel domain integral approach for SIF calculation of 3D cracks with tetrahedral elements and not requiring any structured mesh.It is worth mentioning that SIF of 3D cracks depends on spatial location around the crack edge.For instance,for an internally pressurized rectangular crack,SIF along the length is higher than that along the width.According to Kassir and Sih(1966),a basic characteristic of any 3D crack problem is the fact that the state of stress in a normal plane near a smooth crack front is essentially a plane-strain one.Therefore,for a rectangular crack internally pressurized,SIF is the highest along its length and it can be determined as

wherePHis the internal pressure;ais the half-length of the fracture;ARis the fracture aspect ratio;and the coefficientsf1,f2andf3are defined as 0.5415,1.8086 and 0.6943,respectively.

For almost all of the 3D crack problems,with respect to analytical solutions,SIFs were proposed for single fractures.Therefore,there seems to be lack of enough knowledge about the SIF values for the cases where multiple fractures exist in the medium.In hydraulic fracturing of unconventional reservoirs,due to the existence of many 3D fractures in the medium,the SIF change of multiple fractures placed between parallel lateral wells should be studied.Any fracturing scenario having a higher ratio of SIF with respect to the case of a single fracture in a standalone well can be considered as a fracturing technique with higher propagation potential in the target zone.Study of SIF behavior for different fracturing scenarios can be considered as a simultaneous tool required foroptimization ofhydraulic fracturestogetherwith SSE.Employing these two tools,the spacing and generally the pattern of the fracturing can be designed.

Table 1Input variable range for the numerical simulation.

Regarding the SIF determination of 3D fractures,it is a challenging procedure to obtain acceptable values of SIF using FE method,since a very specific modeling procedure is required for this purpose.For the large number of numerical simulations required for different fracturing patterns(300 models;see Table 1),we tried to use an efficient way for SIFanalysis.Therefore,the ratios of stresses in elements around the crack tip with the same sizes and shapes at equal distances close to the tip are calculated for different scenarios of fracturing compared to that of a single hydraulic fracture.Using this method,any increase or decrease in SIF for any proposed fracturing scenario is reported compared to the case of a single fracture without dealing with the absolute values of SIFs.Considering the results of the current analyses,engineers can investigate whether or not the SIFs of the fractures in a fracturing scenario is changed with respect to a single fracture in an infinite domain qualitatively and quantitatively.As a result,any fracturing scenario can be checked using this methodology in addition to the SSE in order to understand whether there is any fracture trapped or deviated from its straight path.

2.Methodology and conventions

In this work,in order to study the effect of fracture geometry and different fracturing patterns on the SIF of the fractures,different aspect ratios,and multistage and simultaneous fracturing schemes are simulated using FE-based software,ABAQUS CAE 6.12.Modeling of 3D hydraulic fractures is done by assuming hydraulic fractures contained inside the target zone,perpendicular to the wellbore.Relative placement of the hydraulic fracture with the target zone and wellbore,and its geometry definition are shown in Fig. 1,in addition to the hydraulic fracture with the SIFs under study.SIF along the height of a hydraulic fracture is calledSIFHwhich causes propagation in the horizontal direction,while SIF along the length of a hydraulic fracture is calledSIFLwhich causes propagation in the vertical direction.

In addition to the SIF change,aperture of hydraulic fractures mayalso be influenced by fracturing pattern.According to Sneddon and Elliot(1946),aperture of a hydraulic fracture in toughnessdominated regime along the length can be represented by an elliptical function as

where νis the Poisson’s ratio,Enis the Young’s modulus of net play,w（x）is the half aperture,andwmaxis the maximum half aperture(located at the center of the fracture(x=0)).From Eq.(5a),we observe that the displacement of crack tips(atx=c)is zero and the displacement of the edges increases from the tips to the center of the fracture where the maximum displacement occurs.When the maximum value of half aperture is known,by substituting it into Eq.(5a),the aperture values along the length and height can be determined.Therefore,having the maximum half aperture located in the center of the fracture,one can have the aperture distribution on the whole fracture surface.In this study,the term aperture change is the ratioof this maximumhalf aperturefordifferent cases with respect to the single fracture in a standalone well.Fig. 2 shows the scenarios considered for different hydraulic fracturing patterns.

It is evident in Fig. 2 that four different scenarios have been considered for investigation of propagation potential and aperture of hydraulic fractures in the target zone.Scenario 1 is the basic scenario inwhich we have a single stage fracture from a standalone well.Scenario 2 shows multistage hydraulic fracturing in a standalone well.Scenario 3 considers the effect of simultaneous single stage fracturing of the medium between two parallel wells.Finally,in Scenario 4,both of the fracturing strategies,i.e.simultaneous and multistage hydraulic fracturing,are considered.In order to study the change of propagation potential and aperture variation of hydraulic fractures as a result of fracturing pattern,SIFs and apertures of fractures in Scenarios 2-4 arecomparedwith those in Scenario 1.

Fig. 1.(a)Position of the hydraulic fracture with respect to the target zone and wellbore associated with its geometry definition,and(b)Propagation direction according to the SIFs along the length and height of the fracture.

Fig. 2.Different fracturing techniques/patterns in this study.Lpis the half-spacing between multistage fractures,and Lsis the distance between fracture tips.

Changing the fracture geometry by its aspect ratio,the spacing between fractures in multistage fracturing,and the distance between the meeting fractures in simultaneous fracturing scheme,one can study the changes of SIF and aperture of the fractures compared to Scenario 1 which is the basic scenario as a single fracture in an infinite medium.Using this method,the effect of these dimensional variables on the SIFand the aperture of fractures can be quantif i ed.Moreover,the qualitative study for the effect of offset between the meeting fractures on the SIF change is conducted as well.The term offset means the distance between two planes in which hydraulic fractures are located in a simultaneous fracturing scheme.The propagation potential and aperture of hydraulicfracturesare investigated considering the following assumptions:

(1)Constant pressure insidehydraulicfractures has been assumed in all the numerical models.

(2)In modeling of media containing more than one hydraulic fracture,similar hydraulicpressuresand aspectratios together with the same mesh type and size have been assumed for all the hydraulic fractures.

(3)Modeling has been done in a completely elastic medium without considering any plasticity constitutive law.

(4)Hydraulic fracture has been assumed as a stationary crack without considering any propagation.

(5)The considered hydraulic fracturing regimes are assumed as toughness-dominated rather than viscosity-dominated(see Detournay,2004;Bao et al.,2017).

(6)The aspect ratio is defined as the ratio of height to length of the crack and is always equal to or less than unity in this study.

(7)The geometry of the hydraulic fracture has been assumed as a square(aspect ratio of unity)and as rectangles(aspect ratios less than unity).

(8)All the modeling has been performed in three dimensions.

(9)In the scenario of having simultaneous fractures,the distance between fracture tips is controlled by changing the distance between the wells but not the fracture geometry or its aspect ratio.

(10)Symmetry has been assumed for fractures for more efficient simulation.

(11)Due to the low permeability of the reservoir,fluid diffusion time-scale is assumed much longer than the fracturing timescale and only undrained response is considered herein.

In this study,a range of fracture spacing and distance has been assumed by which interaction between neighboring fractures is observed for all the simulations.Input variables and their ranges are listed in Table 1:five different aspect ratios with interval of 0.2,ten different fracture spacings in multistage fracturing,and six different fracture distances in simultaneous fracturing.

Table 1 shows that a number of numerical models(300 models for the quantitative simulation)are required to be built and analyzed to predict the SIF changes and aperture variation in different fracturing scenarios.Fig. 3 shows model geometry,assigned mesh,and symmetry that have been used.It is evident thatonly one-eighth of the hydraulic fracture has been simulated in three dimensions.

The methodology of estimating the SIF ratio of any hydraulic fracturing scheme over that of a single hydraulic fracture in an infinitedomain is explained herein.It is known that stressat a point in the vicinity of the crack tip in the plane where the fracture is located inside(r=R,θ=0?）is related to the SIF using the following equation:

whereRis the distance from tip;KIis linearly related to the internal pressure(PH),which is directly related to a function of the geometry of the fracture,and a function incorporating the effect of boundaries on the fracture.In this way,having the ratio of stress for a fracturing scenario under study over that of the basic Scenario 1,one can have the ratio of the SIF for the fracturing technique over that of the basic Scenario 1.Therefore,we have

Fig. 3.(a)Numerical mesh and(b)model configuration for simulation of one-eighth of a hydraulic fracture.

Fig. 4.Stress variations along the fracture edges for AR=1 and 0.2.

Eq.(7)shows the direct relationship between the ratio of stresses along fracture edge and the ratio of SIF for the scenario under study over that of the basic Scenario 1.Therefore,stresses along the length and height of the fracture,showing an exponential behavior(see Fig. 4),can be indicative of SIF of the fracture edge.One important point in this regard is the singularity of stress on the crack edge which means that the comparison of stresses should be done along a structured mesh with the same type and size.Only under this circumstance,the stresses are comparable due to the precision of stress calculation and the same distance to the tip for both of the scenarios.The other important point is that SIF changes along the length and height of the fracture.This means that the SIF of a rectangular crack along every point on the half-length and halfheight of the fractures is different.Hence,in order to have a representing value of the SIF for the length or height of a fracture,a fi tting function is used to be fi tted on the stresses along the length of the fracture(σzz-x)and the other on stresses along the height of the fracture(σzz-y).The ratio of fi tting function coefficients for the designed scenario over a single fracture is used as a representative of SIF change(see solid lines in Fig. 4).The proposed functions,which give satisfactory fi t on the numerical stress values,are given by

where σzz0is the normal stress at the corners;（σzz）Land （σzz）Hare the normal stresses along the length and height of the fracture,respectively;xandyaxes originate from the corner toward the fracture length and height;bis the half height of fracture;andaHiandaLi(i=1 and 2)are the coefficients of the function.Observing the behavior of the proposed function,it is revealed that stress change is controlled bya1rather thana2.This is because the exponential part is directly multiplied byaH1andaL1,whileaH2andaL2are only the components of the exponential part with negative sign.Therefore,aH1andaL1were considered as sufficient variables for showing the SIF change along the edge of a rectangular hydraulic fracture.Fig. 4 shows a typical example of normal stress change along the crack edge.Considering the representative coefficient of the function(aL1,aH1)proposed in Eqs.(8a)and(8b),one can make a good comparison between the SIF of different fracturing scenarios and the case of a single fracture in an infinite domain.

Fig. 5.Stress validation in the vertical direction away from the fracture center.(a)Plane-strain crack,and(b)Penny-shaped crack.

Fig. 6.SIF changes along the(a)height and(b)length of the fracture as a result of having multistage hydraulic fractures from a standalone well.

It is evident from Fig. 4(solid lines are predictions and dotted lines are numerical values from the software)that the behavior of stress along crack length or height has satisfactorily been predicted using the proposed Eq.(8).It is also seen that normal stress is the highest in the middle point of the crack length and it is reduced to the minimum value at the crack corner.In addition,stress change along the height and length of fractures with aspect ratio of unity is similar as expected(aL1=aH1),but for cracks with aspect ratios less than unity,stress decreases in both the length and height of the fracture.The magnitude of decrease is more severe in the fracture height compared to that in the length.Considering Eq.(6),one can see that higher stress means higher SIF which leads to higher propagation potential.Therefore,as it is seen in Fig. 4,hydraulic fractures with aspect ratios less than unity have more propagation potential in up/down directions than that in left/right directions(SIFL＞SIFH).

3.verification of numerical simulation with analytical solutions

In order to perform a valid numerical simulation,it is first required to numerically model some specific crack problems using the proposed methodology and make a comparison between available analytical solutions and the obtained numerical simulations.Any crack can be specif i ed in space by three geometrical parameters:length,width(called aperture),and height.The ratio of any of these three parameters over the other can be called aspect ratio of the fracture.In this paper,the ratio of height to length is called aspect ratio(AR=height/length).In this section,two cracks with different geometries,i.e.a plane-strain crack with infinite height(AR→+∞),and a penny-shaped crack(AR=1),both with internal pressure,are numerically simulated and the induced stresses around the crack are compared with those of the analytical solutions.Sneddon and Elliot(1946)reported the analytical solution for stress around a crack in an infinite 2D medium with internal pressure.The formulation for penny-shaped cracks was also derived by Sneddon(1946).

The validation approach described here compares the horizontal and vertical stresses around the simulated crack with the analytical results presented by Sneddon and Elliot(1946).Fig. 5 shows the horizontal and vertical stresses along a line perpendicular to the face of the hydraulic fracture.Fig. 5 demonstrates good agreement between the analytical solutions and numerical results for plane-strain and penny-shaped cracks.Therefore,the numerical modeling strategy adopted here can be used for modeling of hydraulic fractures.

4.Effect of multistage fracturing on SIF and aperture in standalone wells

Due to the fact that multistage fracturing is one of the effective techniques for fracturing of unconventional reservoirs,in this section,we investigated the effect of adjacent hydraulic fractures from a standalone well(Scenario 2;Fig. 2)compared to the case of having a single fracture in an infinite medium(Scenario 1;Fig. 2)on the SIF of each scenario.To do so,parallel hydraulic fractures with varying spacing and aspect ratio are assumed,and the SIF change with all the available configurations is studied along the length and height of the fractures.Fig. 6 shows the behaviors ofSIFHandSIFLof the hydraulic fractures of multistage fracturing technique with respect to a single stage fracture,in whichSIF（H，L）MandSIF（H，L）Ostand for SIFs for the height and length of multistage fractures and the single stage fractures in an infinite medium,respectively.

It can be seen from Fig. 6 that the SIF theoretically decreases to 0.21 of its original value when the spacing between the fractures is as low as 0.17c.This means that too closely-spaced fractures causethe SIF to decrease significantly.Therefore,this fracturing pattern may not be suitable and thus a much higher fracturing pressure is required for the fracture to propagate.In addition,when this fracture is located in the shadow region of the first fracture,it may deviate from its original path.It is also seen that the aspect ratio of the hydraulic fracture plays an important role in SIF change of the multistage fractures.In fact,going back to the original state(SIF（H，L）M/SIF（H，L）O→1)occurs in a shorter spacing for fractures with lower aspect ratios.Moreover,comparing the changes ofSIFHandSIFLof hydraulic fractures,it is induced thatSIFLis more influenced by multistage fracturing thanSIFH(withARconvention;see Section 3).This causes the propagation potential to decrease more in vertical directions compared to that in horizontal directions,which results the fracture to stay in the net play and be contained rather than moving inside the bounding layers.

Table 2Coefficients of the function predicting SIF reduction with distance between multistage fractures.

Fig. 7.SIFHratio of simultaneous fractures with respect to standalone fractures.

Table 3Function coefficients for predicting SIFHratio of simultaneous over standalone fracturing.

In order to have a quantif i ed SIF change in multistage fracturing technique,based on the behavior of the numerical results and the proposed technique for SIF ratio calculation,the following equation is proposed:

5.Effect of simultaneous fracturing on SIF in parallel wells

In this section,two horizontal wells are assumed to be placed parallel to each other and only one hydraulic fracture for each well is considered(Scenario 3;Fig. 2).It is also assumed that both of the hydraulic fracture faces align on a single plane without any offset.The distance between fracture tips is changed by assuming different distances between the two lateral wells.It was observed that the meeting edges are influenced by each other and other edges do notshowanychange.This means thatas theverticaledges(height)of simultaneous fractures are meeting each other,SIFHis much affected(increased)by the simultaneous fracturing.This is the reason thatSIFLdoes not show significant change by this fracturing pattern.Therefore,SIFHofsimultaneousfracturesis compared to that of a single stage fracture from a standalone well(Scenario 1;Fig. 2).The results ofSIFHchange are shown in Fig. 7 for different aspect ratios.As it can be seen from Fig. 7,SIFHof simultaneous fractures is controlled by two key variables:distance between the tips and aspect ratio.

Fig. 8.Effects of distance(Ls)and offset(Lo)between fracture tips on the SIF of the fractures.(a)AR=1,and(b)AR=0.4.

Fig. 9.Effects of multistage simultaneous fracturing on(a,b)SIFHand(c,d)SIFLof hydraulic fractures.

In addition,it is observed that aspect ratio plays an important role in thisSIFHincrease in such a way that higher aspect ratio results in higherSIFHincrease.The reason for this observation is that the region of inf l uencing stress field is larger for the case of higher aspect ratios.Based on the behavior of the observed numerical results,this increasing effect can be quantif i ed by fracture aspect ratio and distance between the tips.In order to have a quantifying equation for SIF ratio of simultaneous fractures(Scenario 3)versus single stage fractures(Scenario 1),the following relationship is proposed:

whereis the ratio determining the increasing effect ofthe meeting tips,andnis the coefficient varying with aspect ratio of the fractures.coefficientncan be determined considering Table 3.

Of course,all these values are only valid for single stage simultaneous fractures without having any multistage ones.Considering the numerical results,it is observed that there is no aperture change in simultaneous fracturing scenario compared to the case ofa single fracture.This means that any change in the aperture is merely due to the multistage fracturing.

Table 4Coefficients of the predicting function for SIFHchange as a result of simultaneous multistage fracturing.

The final point regarding simultaneous fracturing between parallel wells is that fractures may propagate in between the wells with some offset between their tips.Consequently,the effect of tip offset between the meeting fractures should also be studied.Fig. 8 depicts the results for aspect ratios of 1 and 0.4.

Fig. 8 shows that the effect of offset is similar to the effect of tip distance.This means thatSIFHraise is reduced by increasing the tip distance and offset.For the case of offset,however,the decrease of theSIFHis smoother by increasing the offset.This means that increasing the distance between the tips removes the effect of simultaneous fracturing in a sharper fashion than the increase of offset.Comparing the results shown in Fig. 8,it is evident that aspect ratio of the fractures also plays the same role,i.e.the lower the aspect ratio of the fracture,the lower the raise inSIFHof the meeting tips.

6.Effect of simultaneous fracturing on SIF of multistage fractures in parallel wells

In simultaneous multistage hydraulic fracturing,effect of meeting fracture tips is mixed with the effect of multistage fracturing.In this section,we intend to investigate the coupled effect of simultaneous multistage hydraulic fracturing by comparing the SIF of this scenario(SIF（H，L）SM)with thatof a singlestagefracture froma standalonewell(SIF（H，L）O).It is clear from the previous sections that multistage fracturing has negative influence on bothSIFLandSIFH(i.e.decreasing effect),while simultaneous fracturing has positive influence onSIFH(i.e.increasing effect).Therefore,the two effects are coupled according to the fracturing configuration whether positive(increasing SIF)or negative(decreasing SIF)effect is observed for the height of the fracture.The coupled effect of simultaneous multistage fracturing on theSIFHandSIFLis shown in Fig. 9.It was observed that SIF shows the same behavior for all aspect ratios,typically for two aspect ratios of 1 and 0.4.The depicted surfaces shown in Fig. 9 are forSIFH(Fig. 9a and b)and forSIFL(Fig. 9c and d)of the fractures.It is observed thatSIFHincreaserelates to the smaller distance between the tips and the larger spacing between multistage fractures.There are alsoLsandLpbounding values,beyond which there is no change of SIF during fracturing.

Table 5Coefficients of the predicting function for SIFLchange as a result of simultaneous multistage fracturing.

Another point which is seen from Fig. 9a and b is that when spacing between multistage fractures is too small,simultaneous fracturing has no influence to keep theSIFHin an acceptable range.This means that multistage fracturing effect dominates the SIF change for too close spacing and causes its substantial reduction,while simultaneous fracturing,even with really low tip distances,cannot remedy this reduction.

Comparing the results shown in Fig. 9c and d with those in Fig. 9a and b,it is revealed that simultaneous fracturing technique has no significant effect on theSIFLof the hydraulic fracture.This is benef i cial for hydraulic fracturing optimization,because by an efficient selection of spacing between the fractures,SIFHis raised as a result of simultaneous fracturing,whileSIFLis reduced by multistage fracturing.This causes the fracture to remain contained in the net play and does not penetrate in the bounding layers.

Similar to the previous sections,it is also important to quantify the SIF change according to the distance between fracture tips and spacing between adjacent fractures in simultaneous multistage fracturing technique.Based on the behaviors of the numerical results,SIF changes can be predicted using the following equation:

7.Conclusions

A comprehensive simulation framework was proposed in order to investigate the interaction between fractures and the influence of closely spaced fractures on SIF.Different scenarios were considered,i.e.specifically fracture aspect ratio,spacing between multistage fractures,and distance between simultaneous fractures.Firstly,it was shown that multistage fracturing can reduce the SIF and aperture of the propagating fractures.The level of this decrease is more severe for higher aspect ratios and shorter spacing between fractures.Secondly,it was observed that the SIF of the meeting hydraulic fractures increases noticeably as a result of simultaneous fracturing.The magnitude of change is again more severe for higher aspect ratios and shorter distances between the tips.It was also observed that the aperture of fractures is not influenced by simultaneous fracturing scheme.Thirdly,it was observed that by having simultaneous multistage fracturing of parallel wells,the increasing effect of simultaneous fracturing is coupled with decreasing effect of multistage fracturing.Decreasing of the SIF and lower propagation potential are seen for propagation in vertical direction,while depending on the spacing(between multistage fractures)and distance(between simultaneous fractures),the resultant effect can be increasing or decreasing for propagation in horizontal direction.Fourthly,it was observed that offset reduces the intensifying effect of simultaneous hydraulic fracturing.All the above-mentioned decreasing and increasing effects on SIF and aperture in different fracturing scenarios were quantif i ed using appropriate fitting equations and the final equations together with their associated coefficients were reported.Using the proposed equations,engineers can predict the SIF change as a result of the designed fracturing pattern.It enables improved planning and placement of productive hydraulic fracture treatments;it offers the potential for considerable cost reductions in completion design and implementation;and it allows for an optimal simultaneous multistage hydraulic fracture treatment that drains larger volumes of the reservoir.

Conflict of interest

The authors wish to confirm that there are no known Conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.

Acknowledgement

The authors would like to thank Mr.Timothy Beard(Manager-ETG Operations,Chesapeake Energy Corporations),and Prof.Arul Britto(Emeritus faculty,University of Cambridge)for their productive advice.Final thanks are also given to Oklahoma Department of Transportation(ODOT)and Oklahoma Transportation Center for their financial support during the course of this study.

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