coupled thm and matrix stability modeling of hydroshearing

Research ArticleCoupled THM and Matrix Stability Modeling of HydroshearingStimulation in a Coupled Fracture-Matrix Hot Volcanic System

Yong Xiao 12 Jianchun Guo 2 HehuaWang13 Lize Lu13

JohnMcLennan4 andMengting Chen5

1China Zhenhua Oil Co Ltd Beijing China2State Key Laboratory on Oil and Gas Reservoir Geology and Exploitation Southwest Petroleum University Chengdu China3Chengdu Northern Petroleum Exploration and Development Technology Co Ltd Chengdu Sichuan China4Energy amp Geoscience Institute University of Utah UT USA5Borehole Operation Branch Office of Sinopec Southwest Petroleum Engineering Co Ltd Sichuan China

Correspondence should be addressed to Jianchun Guo guojianchunvip163com

Received 20 March 2018 Accepted 30 May 2018 Published 3 July 2018

Academic Editor Mohammed Nouari

Copyright copy 2018 Yong Xiao et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A coupled thermal-hydraulic-mechanical (THM) model is developed to simulate the combined effect of fracture fluid flow heattransfer from thematrix to injected fluid and shearing dilation behaviors in a coupled fracture-matrix hot volcanic reservoir systemFluid flows in the fracture are calculated based on the cubic law Heat transfer within the fracture involved is thermal conductionthermal advection and thermal dispersion within the reservoir matrix thermal conduction is the only mode of heat transfer Inview of the expansion of the fracture network deformation and thermal-induced stress model are added to the matrix nodersquos insitu stress environment in each time step to analyze the stability of the matrix A series of results from the coupled THM modelinduced stress andmatrix stability indicate that thermal-induced aperture plays a dominant role near the injection well to enhancethe conductivity of the fracture Away from the injection well the conductivity of the fracture is contributed by shear dilation Theinduced stress has the maximum value at the injection point the deformation-induced stress has large value with smaller affectedrange on the contrary thermal-induced stress has small value with larger affected range Matrix stability simulation results indicatethat the stability of the matrix nodes may be destroyed this mechanism is helpful to create complex fracture networks

1 Introduction

With the increase of energy consumption the explorationand development of oil and gas has gradually shifted from theconventional to unconventional reservoir such as volcanicreservoirs [1 2]The volcanic reservoir has the characteristicsof high temperature gradient and rock mechanical strengthand natural fractures are an important indicator of economicviability The most important consideration for oil and gas orgeothermal energy development in volcanic reservoirs is theuse of hydraulic stimulation technology to establish a fracturenetwork to effectively conduct fluid between injection andproduction wells [3ndash7] In hydraulic fracturing associatedwith oil amp gas or enhanced geothermal energy extractionsignificant differences in surface pumping pressure and theinjectivity index are common even in the same field asshown in Table 1

The purpose of fracturing is to improve the conductivityof natural fractures and possibly extend or develop newfractures [4ndash8] As Table 1 indicates hydraulic fracturingcan include extreme (high or low) injection pressure andoften there are relatively low injection rates with largetotal injected volumes Researchers [9 10] found that twofactors contributed to the difficultly of hydraulic stimulation(1) with the propensity for reactivation of existing faultsand extension or shearing of natural fractures hydraulicstimulation in volcanic reservoirs is sometimes referred toas hydroshearing (2) these stimulation behaviors encom-passed in thermal-hydraulic-mechanical (THM) or thermal-hydraulic-chemical (THC) such as shear dilation and ther-momechanical aperture closure

The injected water flow pattern of the volcanic reservoir ismainly controlled by the network of preexisting and induced

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 3015015 15 pageshttpsdoiorg10115520183015015

2 Mathematical Problems in Engineering

Table1Injectivity

indexdu

ringstimulations

inhydrocarbo

nandgeotherm

algraniticformations

Different

Energy

Develo

pment

Project

Depth

mLitholog

yMaxInj

Rate

m3m

in

Inj

Pressure

MPam

Total

Injection

Volume

m3

Injectivity

Indices

kgsec M

PaRe

ference

OilandGas

Liaohe1

3022

Granite

Gneiss

64

00218

572

162

Liaohe2

4772

55

00145

498

133

Liaohe3

4188

44

00205

480

085

Geothermal

GPK

23600

Granite

36

00036

3000

0462

(Baria200

9)Habanero1

4421

Granite

1800147

2000

010

0[21]

Habanero4

4205

Granite

32

00114

2700

01600

[21]

RRG-9

ST-1

1524

Granite

1000036

303

[22]

Mathematical Problems in Engineering 3

x

zy

Hot Granite Matrix

Natural Fracture x

y

Inj Out

Tf

x= 0

Tf

x= 0

Tm

y= 0

2b

Lm

Lm

Lf

TmTf

Figure 1 Idealized hot matrix-natural fracture system

natural fractures and is affected by changes in conductivityresulting fromheat transfer shear dilation and chemical pro-cesses Many detailed studies of the precipitationdissolutionprocesses and their impact on fracture aperture variation infractured volcanic reservoirs can be found in the literature[11ndash13] The results show in short-term hydroshearing frac-turing the effect of chemical action on the aperture changeis smaller than thermal and mechanical processes so it canbe neglected Therefore when employing hydroshearing inthe volcanic reservoir aperture changes due to shear dilationand thermal stress result in the decrease of net pressure andgrowth of contract area between fracture planes and this inturn affects fluid flows heat transfer and shear behaviors [1415] To explain these behaviors the authors built a modelingstructure that couples thermal hydraulic and mechanicaleffects (THM) and apply this simulator to a hot granitereservoir to examine the importance of this coupling

Because of the complexities of the numerical modelingdeformation and thermal-induced stress are often neglectedand the matrix nodes assume stability and no secondaryfractures form [16] This paper describes a displacementdiscontinuity method [17] to solve the deformation-inducedstress The fracture is considered as a displacement discon-tinuity over a finite-length segment which is treated as theinner boundary condition of the problem [18 19] Afterthe given induced stress the new stress state of the matrixnodes can be updated The new stress state consists of threeparts initial stress deformation-induced stress and thermal-induced stress The new stress state will affect the stability ofthe rockmatrix and help to create complex fracture networks

2 Physical System and Governing Equations

The hot and fractured volcanic oil amp gas reservoir is alwaysconsidered nonpermeability and natural fractures within thematrix are the channels for fluid flow and heat exchangeBased on the simple principle of reducing the extremelycomplex system to an idealized one an idealized system forintegrating hot matrix and natural fracture is established tostudy the fracture deformation and their induced stress Thegeometry of the conceptually idealized model correspondingto a fracture-matrix coupled system is shown in Figure 1

The upstream boundary along the fracture axis represents theinjection well (inlet) and the downstream boundary is theproduction well (outlet)

21 Fracture Fluid Flow Model and Heat Transfer

211 Fracture Fluid Flow Model It is assumed that thefracture aperture varies in space and time and the interfaceof the fracture does not leak-off so fluid flows along thenatural fracture The fracture is assumed to correspond to aparallel-plate system and the laminar flow is valid Withoutconsidering the slip effect the velocity is integrated in thefracture cross section [20] For this fracture system themomentum balance indicates that the average flow velocityper unit length is proportional to the pressure gradient Thefracture fluid flow model is

119906119909 (119909 119905) = 1199082 (119909 119905)12120583 (minus120597119875 (119909 0 119905)120597119909 ) (1)

where u is the average velocity in the fracture per unit length119908 is the average aperture or width of the fracture unit length120583 is the viscosity of the injected fluid which is assumed to bea static value P(x0t) is the fluid pressure within the fracture(y=0) x is the position and t is the time Assuming thatthere is no leak-off and the injected fluid is incompressibleaccording to the continuity equation fluid discharge andchanges per unit height can be written as

119902 (119909 119905) = 119908 (119909 119905) 119906119909 (119909 119905)120597119902 (119909 119905)120597119909 = 0 (2)

where q(x t) is the volumeflowThe fracture deformation andheat transfer are based on the balance of energy and massrespectively The flow equations will be coupled with thefracture constitutive model and thermal-induced aperturechange model

212 Heat Transfer Equations Temperature differences drivethe heat transfer between the hot reservoir matrix andinjected cold water The main mechanisms involved are ther-mal conduction thermal advection and thermal dispersion


us

n

i

dnus

un

n Normal stress

Shear Stressun Aperture

us Shear displacement

dn Dilation angle

Figure 2 Graphs of the fracture aperture caused by shear displacement and dilation

within the fracture conduction-limited thermal transportfrom the reservoirmatrix to the fractureWithin the reservoirmatrix thermal conduction is the only mode of heat transfer[20]

The following general assumptions are used in this sim-ulation for calculating the fracturersquos temperature profile ineach time step (1) specific heat capacities of the matrix andinjected water are not functions of temperature (in otherwords they are static parameters) (2) there is no fluid leak-offform the interface into rockmatrix (ie pressure equilibriumexists between the fracture and the reservoir matrix) Thegoverning equations used in this fracture and the reservoirmatrix heat transfer studies are given as

120588119891119888119891 120597997888rarr119879119891120597119905 = minus120588119891119888119891997888rarr119906nabla997888rarr119879119891 + 119870119891119890119891119891nabla2997888rarr119879119891

+ 120572119891ℎ119898119891 nabla997888rarr119879119898100381610038161003816100381610038161003816119910=119887120588119904119888119904 120597

997888rarr119879119898120597119905 = 119870119898119890119891119891nabla2997888rarr119879119898 minus 120572119891ℎ119898119891 nabla997888rarr119879119898100381610038161003816100381610038161003816119910=119887

(3)

where t is the time Tm and Tf are temperature in the matrixand fracture separately 120588119904 and 120588119891 are rock and water densitycs and cf are rock and water heat capacity Kmeff is the rockthermal conductivity

22 Fracture Constitutive Modeland Deformation-Induced Stress

221 Fracture Constitutive Model The rough surface of anatural fracture is always undulating During the hydroshear-ing stimulation the effective normal stress of the fracturesurface will continue to decrease and the permanent sheardisplacement will be generated The interface of the naturalfracture will no longer completely overlap under normalstress and the dilation aperture 119908119889 will increase because ofshear displacement as shown in Figure 2

In 1985 Barton [23] summarized a nonlinear peakshear strength equation that was based on the joint surface

roughness coefficient and the compressive strength (as repre-sented by JRC and JCS)

120591 = 120590119899 tan [120601119903 + 119869119877119862 log10 (119869119862119878120590119899 )] (4)

where 0r is the internal friction angle the angle within thebrackets is called the friction angle or residual friction angleThe latter term 119869119877119862 log10 (119869119862119878120590119899) is called the dilation anglewhich is closely related to the normal aperture of the fracturesurface The simplest relation between shear strain and stressbefore peak strength can be described in a linear equation as

120591 = 119896119904119906119904 0 le 119906119904 le 119906119901119904 (5)

119896119904 = 120590119899 tan [120601119903 + 119869119877119862119899 log10 (119869119862119878119899120590119899)](119871119899500) [119869119877119862119899119871119899]0330 le 119906119904 le 119906119901119904

(6)

where ks is the shear stiffness 119906119904119901 is shear displacement atpeak strength and Ln is the length of the fracture sampleDilation is defined as a function of shear displacement andtangent of the dynamic dilation angle the equation is shownas

119889119906119899 = tan (119889119899) 119889119906119904 = tan (119889119899) 1119896119904 119889120591 (7)

where dn is the dynamic dilation angle Barton and Choubey[24] estimated the value of the mobilized dilation using thefollowing empirically derived equation

119889119899minus119889119910119899119886119898119894119888 = 12119869119877119862119899minus119889119910119899119886119898119894119888 log10 (1198691198621198781198991205901015840119899) (8)

222 Deformation-Induced Stress The elastic displacementdiscontinuity method (DDM) is an indirect boundary ele-ment method to cope those problems involving pure elas-tic nonporous media containing discontinuous surfaces orthin fractures [17] The elastic DDM is based on analyticalsolution for the infinite two-dimensional homogeneous and


y

x

(x1 y1)

(x y)2aj

(xjyj)

(xjyj)

(xn yn)

yx

j

Figure 3 The local coordinate distribution after the discretion of natural fracture boundary

isotropic elastic nonporous medium containing a finite smallthin fracture with constant normal and shear displacementdiscontinuities For an infinite elastic nonporous mediumthe fractures are divided into N elemental segments with thedisplacement in each segment assumed to have a constantdiscontinuity At any point (x y) the influence of displace-ment discontinuities from all fractures in the system can beobtained by summing the effects of all N elements using thefundamental analytical solutions The fracture length is 2a (ais the half length of fracture segment) and its center is locatedat (0 0) as shown in Figure 3

The deformation-induced stress in the field by the shearand dilation displacement of jth fracture unit is

119895120590119909119909 = 2119866119863119895119904(2119891xy + 119910119891xyy) + 2119866119863119895

119899(119891yy + 119910119891yyy)

119895120590119910119910 = 2119866119863119895119904(minus119910119891xyy) + 2119866119863119895

119899(119891yy minus 119910119891yyy)

119895120590119909119910 = 2119866119863119895119904(119891yy + 119910119891xyy) + 2119866119863119895

119899(minus119910119891xyy)

(9)

where

119891 (119909 119910)= minus 14120587 (1 minus ]) [119910(arctan 119910119909 minus 119886 minus arctan

119910119909 + 119886)minus (119909 minus 119886) lnradic(119909 minus 119886)2 + 1199102+ (119909 + 119886) lnradic(119909 + 119886)2 + 1199102]

(10)

where the field point (119909 y) must transform to the local jthfracture unit coordinate (119909 119910) The (xj yj) is the midpoint ofthe jth fracture unit

119909 = (119909 minus 119909j) cos120573j + (119910 minus 119910j) sin120573j119910 = minus (119909 minus 119909j) sin120573j + (119910 minus 119910j) cos120573j (11)

The deformation-induced stress is approximated as thesum of the all the displacement fracture unit as shown in thefollowing

120590119909119909 = 119899sum119895=1

119895120590119909119909120590119910119910 = 119899sum

119895=1

119895120590119910119910120590119909119910 = 119899sum

119895=1

119895120590119909119910

(12)

23 Thermal-Induced Aperture and Stress Model During thehydroshearing stimulation process a large heat transfers fromthe reservoir matrix into the injected cold fluidThe tempera-ture change causes thermal-induced stress and displacementwithin the rock and fracture The thermal-induced stressmodel under the condition of the continuous heat source isdeduced by using Greenrsquos function [25]

120590119894119895 (119909 119910 119904)= minus119876120588119891119888119891 int119871119891

0119904120590119894119895119888119904 (119877 119904) 120597 (1199091015840 0 119904)

1205971199091015840 1198891199091015840 (13)

where

119888119904119903119903 = minus1198641205721198794120587 (1 minus V) 1198701198981198901198911198911119904 [ 21205852 minus 1198702 (120585) + 1198700 (120585)]

119888119904120579120579 = minus1198641205721198794120587 (1 minus V) 1198701198981198901198911198911119904 [ 21205852 minus 1198702 (120585) minus 1198700 (120585)]

120585 = 119877radic 120588119898119888119898119904119870119898119890119891119891

(14)

The displacement of the granite matrix can be given bythe following Navier equation [20]


119866nabla2119906119910 (119909 119910 119905) + 1198661 minus 2Vnabla [nabla sdot 119906119910 (119909 119910 119905)]= 2119866 (1 + V)1 minus 2V 120572119879nabla119879 (119909 119910 119905)

(15)

where G is the shear modulus v is the Poissonrsquos ratio and uyis the displacement of the fracture to the normal directionIntegrating (15) from the fracture surface to infinity theaperture change of the fracture surface is shown as

minus119906119910 (119909 0 119905) = (1 + V) 1205721198791 minus Vintinfin0

Δ119879 (119909 119910 119905) 119889119910Δ119879 (119909 119910 119905) = 119879 (119909 119910 119905) minus 119879infin

(16)

where Δ119879 is the temperature difference Differentiating (16)with respect to time and using the heat expression for heatconduction in the rock matrix

120597119906119910120597119905 = 12 120597119908 (119909 119905)120597119905= (1 + V) 120572119879119870119898119890119891119891(1 minus V) 120588119898119888119898

120597Δ119879 (119909 119910 119905)120597119910100381610038161003816100381610038161003816100381610038161003816119910=0

(17)

TheBodvarsson analytical solution [26] was used to avoidthe numerical oscillations and uncertainty in numericallycalculated heat flux at the interface

1198791198980 minus 119879 (119909 119910 119905)1198791198980 minus 119879119894119899= 119890119891119888[[

[radic119870119898119890119891119891120588119898119888119898119876119888119891120588119891radic119905 119909 + radic 120588119898119888119898119870119898119890119891119891119905

1199102]]](18)

Assuming that the initial temperature of the granite rockis 1198791198980 and 119879119894119899 is the injection fluid temperature and 119862119908 is thewater specific heat Differentiating (18) with respect to y

120597119879 (119909 119910 119905)120597119910100381610038161003816100381610038161003816100381610038161003816119910=0 =

2Δ1198791198602radic120587 exp [minus (1198601) 2] (19)

Substitution (19) into (17) yields

120597119908 (119909 119905)120597119905= 2 (1 + V) 120572119879119870119898119890119891119891(1 minus V) 120588119898119888119898 (2Δ1198791198602radic120587 exp [minus (1198601)2])

(20)

where A1 and A2 are

1198601 = 119909radic119870119898119890119891119891120588119898119888119898119876120588119891119888119891radic119905

1198602 = radic1205881198981198881198982radic119870119898119890119891119891119905

(21)

Table 2 Parameters used in numerical calculations

Parameters ValueInjected water density (kgm3) 1000Injected water heat capacity (JKgK) 4180Hot dry rock fracture roughness 15Hot dry rock Youngrsquos modulus (Pa) 4E10Hot dry rock Poissonrsquos ratio 028Hot dry rock density (kgm3) 2650Hot dry rock initial temperature (∘C) 200Hot dry rock specific heat capacity (JKgK) 1070Hot dry rock thermal conductivity(JsmK) 260Fracture begin shear expansion coefficient 03

24 Rock Matrix Stability Model The induced stress will begenerated through the deformation of single fracture whichwill lead to redistribution of the stress state around the rockmatrix As a result the rock matrix nodes and unconnectednatural fractures may fail under the new stress environmentand the underground fracture network of hot rock is formedby the communication between the natural fractures

Criterion of rock matrix failure is

1205901 ge 1205903 + 2 (119888 + 1205903 tan120593119887)(1 minus tan120593119887 cot120572) sin 2120572 (22)

Criterion of unconnected fracture failure is

1205901 ge 1205903 + 21205903 tan120593119887(1 minus tan120593119887 cot120572) sin 2120572 (23)

where 1205901 1205903 are maximum andminimum horizontal stress cis rock cohesion MPa 120593119887 is basic friction angle 120572 is the anglebetween maximum horizontal stress and effective normalstress

3 Numerical Solution and Verification

31 Model Parameters and Boundary Condition The hightemperature and fractured granite reservoir system is con-sidered two sets of coupled partial differential equations forthe natural fracture and the matrix All the various inputparameters used in the simulations are reported in Table 2

The problem boundary condition of the fracture is pre-scribed along with the fracture direction and the rock matrixboundary is specified at the interface of the fracture-matrixThe initial and boundary conditions are (24)sim(26) for therock matrix and the within natural fracture respectively are

119879119891 (119909 0 0) = 119879119898 (119909 119910 0) = 1198790119908 (119909 0 0) = 1199080119875119891 (119909 0 0) = 119875119900119906119905

(24)

119879119891 (0 0 119905) = 119879119894119899119902 (0 0 119905) = 119902119894119899 (25)


1

3

4

N

5

2

Matrix

Interface

Central fracture

axisNatural fracture nodesGranite matrix nodes

Figure 4 Discretization of the solution domain satisfying both the cell-based finite volume and the nodal-based finite-difference schemes[15]

12059711987911989812059711991010038161003816100381610038161003816100381610038161003816(119909119871119898119905) = 0

12059711987911989812059711990910038161003816100381610038161003816100381610038161003816(0119905) =

12059711987911989812059711990910038161003816100381610038161003816100381610038161003816(119871119891119905) = 0

(26)

32 Numerical Solution In this study the system is solvednumerically using a second-order central-difference finite-difference scheme The solution is iterated in each time stepto satisfy the continuity at the matrix-fracture interface Thegrid size in the fracture is maintained uniform whereas anonuniform size is adopted in the rockmatrix it is to be notedthat all the fracture andmatrix grid size does not change withtime shown as in Figure 4

The heat transfer model is used to calculate the tem-perature profile through the coupling of the fracture andthe matrix The pressure distribution within the fractureis updated from the thermal-induced aperture change Thefracture deformation due to the effective pressure changewill be analyzed through the fracture constitutive modelThe coupling between thermal deformation and hydroflowis iterated at each time step The convergence criterion forheat transfer is that the temperature difference is less than001K The convergence criterion for THM coupling is thatthe aperture difference is less than 000001mm After theTHM coupling the new stress environment of each matrixnodes is updated from the calculation of induced stressThe stability analysis of matrix nodes is also completed Thecoupling between THM induced stress and matrix stabilityis illustrated in Figure 5

33 Verification As previously described in this paper theshear displacement and dilation behavior of a single fractureare comparedwith a numerical solution as shown in Figure 6

It is to be noted that the prepeak behavior is not elastoplasticand it is simply a nonlinear (prepeak) model The bottomof the rock is fixed in all three directions and the topdisplacement will be applied in one of the shear directionsIn this comparative analysis the confining normal stress is2MPa the jointrsquos roughness is 15 and the jointrsquos compressivestrength is 150MPa and other date has been shown in Table 2The results indicate that the lab test and the numerical modelagree very well

During the verification of the heat transfer model thetemperature in both the rock matrix and natural fracture isa relative temperature which is defined as the ratio of currenttemperature to the initial reservoir temperature The initialtemperature of the matrix is 200∘C and the injection wateris 20∘C The water velocity between the inlet and outlet ismaintained at a constant value of 50 m per day and the otherthermal parameters of the watermatrix are given in Table 2The simulation results of the heat transfer and thermal-induced aperture for verification are shown in Figures 7 and 8All comparative studies have shown that the numericalmodeland other researcherrsquos results are in good agreement

4 Case Study and Matrix Stability Analysis

41 Hydroshearing Stimulation and Pressure Fitting RRG-9ST1 well is located in Cassia County southwestern IdahoThe open hole section of the well from 5551 to 5900 ftMD consists of granite and minor diabase The tested initialminimum horizontal stress is 22MPa and the maximumis 32MPa [27 28] The hydroshearing stimulation targetshowed evidence of more than eighty natural fractures andthe maximum temperature range from 130∘C to 150∘C Aftercementing a hydroshearing stimulation programwas carriedout in the open hole section in 2013 The objective of the


Model parameters input rock mechanics heattransfer fluid parameters

Heat transfer modelFractureconduction convection

Heat transfer to balance

Matrix

Flow model pressuredistribution

Fracturing constitutive modelUpdate aperture

Rock matrix stability model

coupling

IterationNo

THM model and induced stress coupled

ConvergenceNo Deformation induced

stressYes

Calculate induced stress total aperturetemp and pressure profile

Nexttime

t+t

Figure 5 Coupled thermal-hydraulic-mechanical model and induced stress

0

02

04

06

0

1

2

3

Dila

tion

(mm

)

Shea

r stre

ss (M

Pa)

0 03 06 09 12Shear displacement (mm)

Shear stress - Numerical modelShear stress - Lab Test(Nassir 2013)Dilation - Numerical modelDilation - Lab Test(Nassir 2013)

n

JRC = 15JCS = 150MPan = 2MPa

Figure 6 Comparison between the lab test and the numerical model for shear stress and displacement


00

02

04

06

08

10

0 50 100 150 200 250 300

Relat

ive t

empe

ratu

re in

frac

ture

Fracture length (m)

1 year

5 year

15 year

Dotted lines ndashCheng 2001 solutionSolid lines ndash Numerical models

Figure 7 Comparison the normalized temperature distribution between numerical model and Chengrsquos solution in the fracture

0098

0100

0102

0104

0106

0108

0110

0112

Ther

mal

indu

ced

half

aper

ture

(m)

100 200 300 400 5000Fracture length (m)

Ghassemi2007 solutionNumerical models

Figure 8 Comparison the thermal-induced aperture along the fracture between numerical model and Ghassemirsquos solution

fracturing program is to improve the injectivity of the wellThe entire stimulation can be divided into three stagesunstable pumping stage stable pumping stage and efficientpumping stage In the unstable pumping stage the injectedwater temperature from high (40∘C) decreased to low (12∘C)as shown in Table 3

At the beginning of the fracturing high temperature(40∘C) water was injected into the fracture The permeabilityof the fracture was very low and nearly equal to the initialconductivity while the surface pumping pressure increaseswith the injection rate After unstable pumping stage thepermeability of the natural fracture is greatly improvedthe ground injection pressure is maintained at a constantvalueThe pumping pressure fitting shows that the simulationresults are in good agreement with the well site fracturingdate as shown in Figure 9 The coupled THM model torepresent fracture aperture change and heat transfer workedwell

42 Fracture Aperture and Temperature Profile Having ver-ified the basic coupled fracture-matrix heat transfer modeland fracture deformation equations the total aperture changealong the fracture is computed Both models that affectthe aperture change have been used in this work and theresults are shown in Figure 10 It can be observed thatboth expressions yield similar trends at different pumpingstages Under the combined influence of thermal stress anddecreasing net pressure the fracture aperture increases withthe shearing fracturing time After 241 days of injectionsimulation the change in fracture aperture in RRG-9 ST1well occurs over a short zone near the injection well (theaperture change zone is about 10m) The fracture apertureis at a maximum (003m) near the injection well and thismaximum aperture decreases along the fracture axis as thefluid circulation continues between the inlet and outlet wells

The combined influence of thermal stress and decreasein net pressure induced deformation in a fracture-matrixcoupled system on fracture aperture is illustrated in Figure 11


Table 3 Hydroshearing stimulation schedule of the RRG-9 ST1 well

Hydro-shearing stimulation Injected water temperature ∘C Injection rate m3min Pumping pressure Psi Time day

Unstable pumping stage 40 016-053-099 270-540-810 3512 096 740-520 6

Stable pumping stage 29 05 270 90Efficient pumping stage 45 08-19 260 110

0

04

08

12

16

2

0

100

200

300

400

500

600

700

800

900

Surf

ace p

ress

ure (

Psi)

Time

Surface pressure--simulationSurface pressure--well siteInjection rate

Inje

ctio

n ra

te (m

m

in)

2015

46

2013

56

2013

81

4

2013

11

22

2014

32

2014

61

0

2014

91

8

2014

12

27

Figure 9 Pressure fitting on RRG-9 ST1 well hydroshearing simulation

0000

0005

0010

0015

0020

0025

0030

0035

Tota

l ape

rtur

e (m

)

01 1 10 100Fracture length (m)

Unstable pumping stage-High temperature waterUnstable pumping stage-Low temperature waterStable pumping stageEfficient pumping stage

Figure 10 Change in fracture total aperture along the fracture due to coupling of thermal and mechanical

Two distinct regimes in fracture aperture variation can beobserved along the fracture between the inlet and out-let wells The first regime occurs near the injection wellabout 10 meters the contribution of fracture conductivity

is mainly from the thermal-induced aperture The secondregime away from the injection well the conductivity ofthe fracture is dominated by shear dilation Therefore thebest way to enhance the conductivity of the underground


Thermal induced apertureDilation aperture

1 10 10001Fracture length (m)

0001

0010

0100

1000

10000

Aper

ture

(mm

)

Figure 11 Change in thermal-induced aperture and dilation aperture along the fracture

0

30

60

90

120

150



Tem

pera

ture

(∘

)

Figure 12 Change in temperature profile along the fracture indifferent pumping stage

hundreds of metersrsquo fracture area is the using of sheardilation

The temperature profiles along the fracture axis afterthree different pumping stages are illustrated in Figure 12The plot of the temperature profile is very sharp near theinjection area which indicates a large heat transfer from thereservoir into the fracture fluid During long-term fracturingit requires a greater distance from the injection well for thefluid to reach thematrix initial temperature In this case after241 daysrsquo injection the temperature of the circulating fluidnearly reaches the matrix initial temperature at 20 m fromthe injection well That is why the thermal-induced aperturechange regime is very small It is indirectly proved that thehot reservoir has a strong thermal energy mining capacity

43 Matrix Stability and Fracture Network Expansion Afterthe calculation of fracture total aperture and heat transfer

in each time step the deformation-induced stress andthermal-induced stress are calculated Figure 13 shows thedeformation-induced stress inminimum andmaximumhor-izontal stress direction The deformation-induced minimumhorizontal stress and maximum horizontal stress have asimilar change trend and have the maximum positive valueat the injection point In other words the stress state ofthe matrix nodes is increased In this case after 241 daysrsquoinjection the maximum value of the deformation-inducedstress in minimum and maximum horizontal stress directionare 35 MPa and 101 MPa separately The maximum valuequickly reduced to a stable value in a very short range nearthe injection well (in this case the affected range is about 1meter)

The thermal-induced stress in both minimum and max-imum horizontal stress directions are negative which meansthat tensile stress is formed near the injection area asshown in Figure 14 In this RRG-9 ST1 well hydroshearingsimulation case after 241 daysrsquo injection the maximum valueof the thermal-induced stress in minimum and maximumhorizontal stress direction are -3 MPa and -45 MPa sep-arately The thermal-induced stress affected range is about2 meters Therefore the deformation-induced stress has alarge value with a smaller affected range on the contrarythermal-induced stress has a small value with a larger affectedrange

After the given of induced stress the new stress stateof the matrixrsquos nodes can be updated The new stress stateconsists of three parts initial stress deformation-inducedstress and thermal-induced stress The new stress state willaffect the stability of the rock matrix In other words whenthe new stress state of the matrix nodes is located above thestrength envelope the stability of the matrix node will bedestroyed Figure 15 shows the relationship between the newstress state and the strength envelope of the matrix (left)the failure modes and their position in different pumpingstages (right) In this simulation case after three-stage frac-turing seven matrix nodes are located above the strengthenvelope With the increase in fracturing pump time thenumber of the failure nodes and the affected range also


Efficient pumping stageStable pumping stageUnstable pumping stage

Inlet

00 05 10 15 20 25 30 Fracture Length (m)

minus5

0

5

10

15

20

25

30

35

Def

orm

atio

n In

duce

d St

ress

(MPa

)

(a) Minimum horizontal stress direction


minus5

15

35

55

75

95

115D

efor

mat

ion

Indu

ced

Stre

ss (M

Pa)


Inlet

(b) Maximum horizontal stress direction

Figure 13 The variation of deformation-induced stress in minimum and maximum horizontal stress direction in different pumping stages

increase This mechanism is helpful to create complex frac-ture networks

5 Conclusions

A coupled THM model is developed to simulate the com-bined effect of fracture fluid flow heat transfer from thematrix to injected fluid and shearing dilation behaviors ina coupled fracture-matrix hot volcanic reservoir system Inview of the expansion of the fracture network deformationand thermal-induced stress models are added to the matrixnodersquos in situ stress environment in each time step to analyzethe stability of the matrix

The coupled THM modeling results indicate that inthese case conditions the change in fracture aperture dueto thermoelastic stress occurs near the injection well thusincreasing the fracture opening and playing a dominant role

Away from the injection well the conductivity of the fractureis contributed by shear dilation Therefore the best way toenhance the conductivity of the underground hundreds ofmetersrsquo fracture area is the using of shear dilation

The induced stress simulation results indicate thatdeformation-induced stress is a compressive stress with apositive and thermal-induced stress is tensile stress with anegative value The induced stress has the maximum valueat the injection point the deformation-induced stress hasa large value with smaller affected range on the contrarythermal-induced stress has a small value with larger affectedrange

The new stress state of thematrix nodes is updated by ini-tial stress deformation-induced stress and thermal-inducedstress The matrix stability simulation results indicate inthis case some matrix nodes are located above the strengthenvelope after hydroshearing stimulation which means the


Unstable pumping stage

Efficient pumping stageStable pumping stage

Inlet

minus35

minus30

minus25

minus20

minus15

minus10

minus05

00

Ther

mal

indu

ced

stres

s (M

Pa)

1 2 3 4 5 60Fracture length (m)


Unstable pumping stageStable pumping stageEfficient pumping stage

Inlet

minus50

minus40

minus30

minus20

minus10

00 Th

erm

al in

duce

d str

ess (

MPa

)



Figure 14 The variation of thermal-induced stress in minimum and maximum horizontal stress direction in different pumping stage

20

40

60

80

100

Max

imum

Hor

izon

tal S

tres

s (M

Pa)

Matrix NodesMatrix Strength Envelop

05

1

15

10 15 20 25 30 35 40 45 50 55Minimum Horizontal Stress (MPa)

00 05 1 15 2

Fracture Length (m)

Dist

ance

Alo

ng th

e Mat

rix

(m)

Failure node aer unstable pumping stageFailure nodes aer stable pumping stageFailure nodes aer hydro-shearing

Figure 15 The relationship between the new stress state and the strength envelope of the matrix (left) the failure modes and their positionin different pumping stages (right)


stability of thematrix nodewill be destroyedThismechanismis helpful to create complex fracture networks

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors would like to acknowledge the support of theNational Science Fund for Distinguished Young Scholars (ID51525404) and Foundation of China Scholarship Council(File no 201608510077)

References

[1] Z Feng J Liu PWang S Chen and Y Tong ldquooil and gas explo-ration field volcanic hydrocarbon reservoirs-Enlightenmentfrom the discovery of the large gas field in Songliao basinrdquoDiqiuWuli Xuebao vol 54 no 2 pp 269ndash279 2011

[2] S A Holditch ldquoThe Increasing Role of Unconventional Reser-voirs in the Future of the Oil and Gas Businessrdquo Journal ofPetroleum Technology vol 55 no 11 pp 34ndash79 2003

[3] Z-Q Feng ldquoVolcanic rocks as prolific gas reservoir A casestudy from the Qingshen gas field in the Songliao Basin NEChinardquoMarine and PetroleumGeology vol 25 no 4-5 pp 416ndash432 2008

[4] J Guo B Luo C Lu J Lai and J Ren ldquoNumerical investigationof hydraulic fracture propagation in a layered reservoir usingthe cohesive zonemethodrdquoEngineering FractureMechanics vol186 pp 195ndash207 2017

[5] J Guo JWang Y Liu Z Chen andH Zhu ldquoAnalytical analysisof fracture conductivity for sparse distribution of proppantpacksrdquo Journal of Geophysics and Engineering vol 14 no 3 pp599ndash610 2017

[6] Y Xiao G Jianchuan and J Mclennan ldquoDeformation andinitiation in Fracture Due To Hydro-Shearing and InducedStress in Enhanced Geothermal Systemrdquo in Paper presented atthe 4th ISRM Young Scholars Symposium on Rock Mechanics2017

[7] Y Xiao G Jianchuan and B Luo ldquoCharacterization of Frac-ture Conductivity of Hydraulic Fracturing in Hot Dry RockExploitationrdquo in Paper presented at the 51st US RockMechanicsGeomechanics Symposium 2017

[8] D Vogler F Amann P Bayer and D Elsworth ldquoPermeabilityEvolution in Natural Fractures Subject to Cyclic Loading andGouge Formationrdquo Rock Mechanics and Rock Engineering vol49 no 9 pp 3463ndash3479 2016

[9] A P Rinaldi J Rutqvist E L Sonnenthal and T T CladouhosldquoCoupled THM Modeling of Hydroshearing Stimulation inTight Fractured Volcanic Rockrdquo Transport in PorousMedia vol108 no 1 pp 131ndash150 2015

[10] J Taron and D Elsworth ldquoThermal-hydrologic-mechanical-chemical processes in the evolution of engineered geothermal

reservoirsrdquo International Journal of Rock Mechanics and MiningSciences vol 46 no 5 pp 855ndash864 2009

[11] J Rutqvist Y-S Wu C-F Tsang and G Bodvarsson ldquoAmodeling approach for analysis of coupled multiphase fluidflow heat transfer and deformation in fractured porous rockrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 39 no 4 pp 429ndash442 2002

[12] J Taron D Elsworth and K-B Min ldquoNumerical simula-tion of thermal-hydrologic-mechanical-chemical processes indeformable fractured porous mediardquo International Journal ofRockMechanics andMining Sciences vol 46 no 5 pp 842ndash8542009

[13] Q Gan and D Elsworth ldquoProduction optimization in fracturedgeothermal reservoirs by coupled discrete fracture networkmodelingrdquo Geothermics vol 62 pp 131ndash142 2016

[14] G Izadi and D Elsworth ldquoThe influence of thermal-hydraulic-mechanical- and chemical effects on the evolution of permeabil-ity seismicity and heat production in geothermal reservoirsrdquoGeothermics vol 53 pp 385ndash395 2015

[15] G S Kumar and A Ghassemi ldquoNumerical modeling ofnon-isothermal quartz dissolutionprecipitation in a coupledfracture-matrix systemrdquoGeothermics vol 34 no 4 pp 411ndash4392005

[16] P Olasolo M C Juarez M P Morales S Damico and IA Liarte ldquoEnhanced geothermal systems (EGS) A reviewrdquoRenewable amp Sustainable Energy Reviews vol 56 pp 133ndash1442015

[17] S L Crouch AM Starfield and F J Rizzo ldquoBoundary elementmethods in solid mechanicsrdquo Journal of Applied Mechanics vol50 p 704 1983

[18] Y Cheng ldquoBoundary element analysis of the stress distributionaround multiple fractures Implications for the spacing of per-foration clusters of hydraulically fractured horizontal wellsrdquo inProceedings of the SPE Eastern Regional Meeting 2009 LimitlessPotential Formidable Challenges pp 267ndash281 usa September2009

[19] Q Tao Numerical modeling of fracture permeability change innaturally fractured reservoirs using a fully coupled displacementdiscontinuity method Texas AM University 2010

[20] A Ghassemi and G Suresh Kumar ldquoChanges in fractureaperture and fluid pressure due to thermal stress and silicadissolutionprecipitation induced by heat extraction from sub-surface rocksrdquo Geothermics vol 360 no 2 pp 115ndash140 2007

[21] E Barbier ldquoGeothermal energy technology and current statusAn overviewrdquo Renewable amp Sustainable Energy Reviews vol 6no 1-2 pp 3ndash65 2002

[22] J Bradford J Moore M Ohren J McLennan W L Osbornand E Majer ldquoRecent Thermal and Hydraulic StimulationResults at Raft Riverrdquo in ID EGS Site Paper presented at theFourtieth Workshop on Geothermal Reservoir EngineeringStanford University Stanford California 2015

[23] N Barton S Bandis and K Bakhtar ldquoStrength deformationand conductivity coupling of rock jointsrdquo in Proceedings of thePaper presented at the International Journal of Rock Mechanicsand Mining Sciences amp Geomechanics Abstracts 1985

[24] N Barton and V Choubey The shear strength of rock joints intheory and practice Rock Mechanics amp Rock Engineering1977

[25] S Tarasovs and A Ghassemi ldquoPropagation of a system ofcracks under thermal stressrdquo in Proceedings of the 45th US RockMechanics Geomechanics Symposium usa June 2011


[26] G Bodvarsson ldquoOn the temperature of water flowing throughfracturesrdquo Journal of Geophysical Research vol 74 no 8 pp1987ndash1992 1969

[27] J Bradford J McLennan J Moore D Glasby D Waters andR Kruwell ldquoRecent developments at the Raft River geothermalfieldrdquo in Proceedings of the Paper presented at the ProceedingsStanford University Stanford California 2013

[28] C Jones J Moore W Teplow and S Craig ldquoGeology andhydrothermal alteration of the Raft River geothermal systemrdquoin Proceedings of the Idaho Paper presented at the ProceedingsThirty-Sixth Workshop on Geothermal Reservoir EngineeringStanford University Stanford California 2011

Hindawiwwwhindawicom Volume 2018

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Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences


Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018


Decision SciencesAdvances in


AnalysisInternational Journal of


Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


Table1Injectivity

indexdu

ringstimulations

inhydrocarbo

nandgeotherm

algraniticformations

Different

Energy

Develo

pment

Project

Depth

mLitholog

yMaxInj

Rate

m3m

in

Inj

Pressure

MPam

Total

Injection

Volume

m3

Injectivity

Indices

kgsec M

PaRe

ference

OilandGas

Liaohe1

3022

Granite

Gneiss

64

00218

572

162

Liaohe2

4772

55

00145

498

133

Liaohe3

4188

44

00205

480

085

Geothermal

GPK

23600

Granite

36

00036

3000

0462

(Baria200

9)Habanero1

4421

Granite

1800147

2000

010

0[21]

Habanero4

4205

Granite

32

00114

2700

01600

[21]

RRG-9

ST-1

1524

Granite

1000036

303

[22]


x

zy

Hot Granite Matrix

Natural Fracture x

y

Inj Out

Tf

x= 0

Tf

x= 0

Tm

y= 0

2b

Lm

Lm

Lf

TmTf









119906119909 (119909 119905) = 1199082 (119909 119905)12120583 (minus120597119875 (119909 0 119905)120597119909 ) (1)


119902 (119909 119905) = 119908 (119909 119905) 119906119909 (119909 119905)120597119902 (119909 119905)120597119909 = 0 (2)




us

n

i

dnus

un

n Normal stress



dn Dilation angle





+ 120572119891ℎ119898119891 nabla997888rarr119879119898100381610038161003816100381610038161003816119910=119887120588119904119888119904 120597


(3)






120591 = 120590119899 tan [120601119903 + 119869119877119862 log10 (119869119862119878120590119899 )] (4)


120591 = 119896119904119906119904 0 le 119906119904 le 119906119901119904 (5)

119896119904 = 120590119899 tan [120601119903 + 119869119877119862119899 log10 (119869119862119878119899120590119899)](119871119899500) [119869119877119862119899119871119899]0330 le 119906119904 le 119906119901119904

(6)


119889119906119899 = tan (119889119899) 119889119906119904 = tan (119889119899) 1119896119904 119889120591 (7)


119889119899minus119889119910119899119886119898119894119888 = 12119869119877119862119899minus119889119910119899119886119898119894119888 log10 (1198691198621198781198991205901015840119899) (8)



y

x

(x1 y1)

(x y)2aj

(xjyj)

(xjyj)

(xn yn)

yx

j




119895120590119909119909 = 2119866119863119895119904(2119891xy + 119910119891xyy) + 2119866119863119895

119899(119891yy + 119910119891yyy)

119895120590119910119910 = 2119866119863119895119904(minus119910119891xyy) + 2119866119863119895

119899(119891yy minus 119910119891yyy)

119895120590119909119910 = 2119866119863119895119904(119891yy + 119910119891xyy) + 2119866119863119895

119899(minus119910119891xyy)

(9)

where



(10)




120590119909119909 = 119899sum119895=1

119895120590119909119909120590119910119910 = 119899sum

119895=1

119895120590119910119910120590119909119910 = 119899sum

119895=1

119895120590119909119910

(12)


120590119894119895 (119909 119910 119904)= minus119876120588119891119888119891 int119871119891

0119904120590119894119895119888119904 (119877 119904) 120597 (1199091015840 0 119904)

1205971199091015840 1198891199091015840 (13)

where



120585 = 119877radic 120588119898119888119898119904119870119898119890119891119891

(14)




(15)



Δ119879 (119909 119910 119905) 119889119910Δ119879 (119909 119910 119905) = 119879 (119909 119910 119905) minus 119879infin

(16)


120597119906119910120597119905 = 12 120597119908 (119909 119905)120597119905= (1 + V) 120572119879119870119898119890119891119891(1 minus V) 120588119898119888119898

120597Δ119879 (119909 119910 119905)120597119910100381610038161003816100381610038161003816100381610038161003816119910=0

(17)


1198791198980 minus 119879 (119909 119910 119905)1198791198980 minus 119879119894119899= 119890119891119888[[


1199102]]](18)


120597119879 (119909 119910 119905)120597119910100381610038161003816100381610038161003816100381610038161003816119910=0 =

2Δ1198791198602radic120587 exp [minus (1198601) 2] (19)



(20)

where A1 and A2 are

1198601 = 119909radic119870119898119890119891119891120588119898119888119898119876120588119891119888119891radic119905

1198602 = radic1205881198981198881198982radic119870119898119890119891119891119905

(21)












119879119891 (119909 0 0) = 119879119898 (119909 119910 0) = 1198790119908 (119909 0 0) = 1199080119875119891 (119909 0 0) = 119875119900119906119905

(24)

119879119891 (0 0 119905) = 119879119894119899119902 (0 0 119905) = 119902119894119899 (25)


1

3

4

N

5

2

Matrix

Interface

Central fracture



12059711987911989812059711991010038161003816100381610038161003816100381610038161003816(119909119871119898119905) = 0

12059711987911989812059711990910038161003816100381610038161003816100381610038161003816(0119905) =

12059711987911989812059711990910038161003816100381610038161003816100381610038161003816(119871119891119905) = 0

(26)












Matrix




coupling

IterationNo



stressYes


Nexttime

t+t


0

02

04

06

0

1

2

3

Dila

tion

(mm

)

Shea

r stre

ss (M

Pa)



n




00

02

04

06

08

10

0 50 100 150 200 250 300

Relat

ive t

empe

ratu

re in

frac

ture

Fracture length (m)

1 year

5 year

15 year



0098

0100

0102

0104

0106

0108

0110

0112

Ther

mal

indu

ced

half

aper

ture

(m)













0

04

08

12

16

2

0

100

200

300

400

500

600

700

800

900

Surf

ace p

ress

ure (

Psi)

Time


Inje

ctio

n ra

te (m

m

in)

2015

46

2013

56

2013

81

4

2013

11

22

2014

32

2014

61

0

2014

91

8

2014

12

27


0000

0005

0010

0015

0020

0025

0030

0035

Tota

l ape

rtur

e (m

)









0001

0010

0100

1000

10000

Aper

ture

(mm

)


0

30

60

90

120

150



Tem

pera

ture

(∘

)










Inlet


minus5

0

5

10

15

20

25

30

35

Def

orm

atio

n In

duce

d St

ress

(MPa

)



minus5

15

35

55

75

95

115D

efor

mat

ion

Indu

ced

Stre

ss (M

Pa)


Inlet




5 Conclusions









Inlet

minus35

minus30

minus25

minus20

minus15

minus10

minus05

00

Ther

mal

indu

ced

stres

s (M

Pa)




Inlet

minus50

minus40

minus30

minus20

minus10

00 Th

erm

al in

duce

d str

ess (

MPa

)




20

40

60

80

100

Max

imum

Hor

izon

tal S

tres

s (M

Pa)


05

1

15


00 05 1 15 2

Fracture Length (m)

Dist

ance

Alo

ng th

e Mat

rix

(m)





Data Availability




Acknowledgments


References






































Journal of












Journal of







Volume 2018






Volume 2018









x

zy

Hot Granite Matrix

Natural Fracture x

y

Inj Out

Tf

x= 0

Tf

x= 0

Tm

y= 0

2b

Lm

Lm

Lf

TmTf









119906119909 (119909 119905) = 1199082 (119909 119905)12120583 (minus120597119875 (119909 0 119905)120597119909 ) (1)


119902 (119909 119905) = 119908 (119909 119905) 119906119909 (119909 119905)120597119902 (119909 119905)120597119909 = 0 (2)




us

n

i

dnus

un

n Normal stress



dn Dilation angle





+ 120572119891ℎ119898119891 nabla997888rarr119879119898100381610038161003816100381610038161003816119910=119887120588119904119888119904 120597


(3)






120591 = 120590119899 tan [120601119903 + 119869119877119862 log10 (119869119862119878120590119899 )] (4)


120591 = 119896119904119906119904 0 le 119906119904 le 119906119901119904 (5)

119896119904 = 120590119899 tan [120601119903 + 119869119877119862119899 log10 (119869119862119878119899120590119899)](119871119899500) [119869119877119862119899119871119899]0330 le 119906119904 le 119906119901119904

(6)


119889119906119899 = tan (119889119899) 119889119906119904 = tan (119889119899) 1119896119904 119889120591 (7)


119889119899minus119889119910119899119886119898119894119888 = 12119869119877119862119899minus119889119910119899119886119898119894119888 log10 (1198691198621198781198991205901015840119899) (8)



y

x

(x1 y1)

(x y)2aj

(xjyj)

(xjyj)

(xn yn)

yx

j




119895120590119909119909 = 2119866119863119895119904(2119891xy + 119910119891xyy) + 2119866119863119895

119899(119891yy + 119910119891yyy)

119895120590119910119910 = 2119866119863119895119904(minus119910119891xyy) + 2119866119863119895

119899(119891yy minus 119910119891yyy)

119895120590119909119910 = 2119866119863119895119904(119891yy + 119910119891xyy) + 2119866119863119895

119899(minus119910119891xyy)

(9)

where



(10)




120590119909119909 = 119899sum119895=1

119895120590119909119909120590119910119910 = 119899sum

119895=1

119895120590119910119910120590119909119910 = 119899sum

119895=1

119895120590119909119910

(12)


120590119894119895 (119909 119910 119904)= minus119876120588119891119888119891 int119871119891

0119904120590119894119895119888119904 (119877 119904) 120597 (1199091015840 0 119904)

1205971199091015840 1198891199091015840 (13)

where



120585 = 119877radic 120588119898119888119898119904119870119898119890119891119891

(14)




(15)



Δ119879 (119909 119910 119905) 119889119910Δ119879 (119909 119910 119905) = 119879 (119909 119910 119905) minus 119879infin

(16)


120597119906119910120597119905 = 12 120597119908 (119909 119905)120597119905= (1 + V) 120572119879119870119898119890119891119891(1 minus V) 120588119898119888119898

120597Δ119879 (119909 119910 119905)120597119910100381610038161003816100381610038161003816100381610038161003816119910=0

(17)


1198791198980 minus 119879 (119909 119910 119905)1198791198980 minus 119879119894119899= 119890119891119888[[


1199102]]](18)


120597119879 (119909 119910 119905)120597119910100381610038161003816100381610038161003816100381610038161003816119910=0 =

2Δ1198791198602radic120587 exp [minus (1198601) 2] (19)



(20)

where A1 and A2 are

1198601 = 119909radic119870119898119890119891119891120588119898119888119898119876120588119891119888119891radic119905

1198602 = radic1205881198981198881198982radic119870119898119890119891119891119905

(21)












119879119891 (119909 0 0) = 119879119898 (119909 119910 0) = 1198790119908 (119909 0 0) = 1199080119875119891 (119909 0 0) = 119875119900119906119905

(24)

119879119891 (0 0 119905) = 119879119894119899119902 (0 0 119905) = 119902119894119899 (25)


1

3

4

N

5

2

Matrix

Interface

Central fracture



12059711987911989812059711991010038161003816100381610038161003816100381610038161003816(119909119871119898119905) = 0

12059711987911989812059711990910038161003816100381610038161003816100381610038161003816(0119905) =

12059711987911989812059711990910038161003816100381610038161003816100381610038161003816(119871119891119905) = 0

(26)












Matrix




coupling

IterationNo



stressYes


Nexttime

t+t


0

02

04

06

0

1

2

3

Dila

tion

(mm

)

Shea

r stre

ss (M

Pa)



n




00

02

04

06

08

10

0 50 100 150 200 250 300

Relat

ive t

empe

ratu

re in

frac

ture

Fracture length (m)

1 year

5 year

15 year



0098

0100

0102

0104

0106

0108

0110

0112

Ther

mal

indu

ced

half

aper

ture

(m)













0

04

08

12

16

2

0

100

200

300

400

500

600

700

800

900

Surf

ace p

ress

ure (

Psi)

Time


Inje

ctio

n ra

te (m

m

in)

2015

46

2013

56

2013

81

4

2013

11

22

2014

32

2014

61

0

2014

91

8

2014

12

27


0000

0005

0010

0015

0020

0025

0030

0035

Tota

l ape

rtur

e (m

)









0001

0010

0100

1000

10000

Aper

ture

(mm

)


0

30

60

90

120

150



Tem

pera

ture

(∘

)










Inlet


minus5

0

5

10

15

20

25

30

35

Def

orm

atio

n In

duce

d St

ress

(MPa

)



minus5

15

35

55

75

95

115D

efor

mat

ion

Indu

ced

Stre

ss (M

Pa)


Inlet




5 Conclusions









Inlet

minus35

minus30

minus25

minus20

minus15

minus10

minus05

00

Ther

mal

indu

ced

stres

s (M

Pa)




Inlet

minus50

minus40

minus30

minus20

minus10

00 Th

erm

al in

duce

d str

ess (

MPa

)




20

40

60

80

100

Max

imum

Hor

izon

tal S

tres

s (M

Pa)


05

1

15


00 05 1 15 2

Fracture Length (m)

Dist

ance

Alo

ng th

e Mat

rix

(m)





Data Availability




Acknowledgments


References






































Journal of












Journal of







Volume 2018






Volume 2018









us

n

i

dnus

un

n Normal stress



dn Dilation angle





+ 120572119891ℎ119898119891 nabla997888rarr119879119898100381610038161003816100381610038161003816119910=119887120588119904119888119904 120597


(3)






120591 = 120590119899 tan [120601119903 + 119869119877119862 log10 (119869119862119878120590119899 )] (4)


120591 = 119896119904119906119904 0 le 119906119904 le 119906119901119904 (5)

119896119904 = 120590119899 tan [120601119903 + 119869119877119862119899 log10 (119869119862119878119899120590119899)](119871119899500) [119869119877119862119899119871119899]0330 le 119906119904 le 119906119901119904

(6)


119889119906119899 = tan (119889119899) 119889119906119904 = tan (119889119899) 1119896119904 119889120591 (7)


119889119899minus119889119910119899119886119898119894119888 = 12119869119877119862119899minus119889119910119899119886119898119894119888 log10 (1198691198621198781198991205901015840119899) (8)



y

x

(x1 y1)

(x y)2aj

(xjyj)

(xjyj)

(xn yn)

yx

j




119895120590119909119909 = 2119866119863119895119904(2119891xy + 119910119891xyy) + 2119866119863119895

119899(119891yy + 119910119891yyy)

119895120590119910119910 = 2119866119863119895119904(minus119910119891xyy) + 2119866119863119895

119899(119891yy minus 119910119891yyy)

119895120590119909119910 = 2119866119863119895119904(119891yy + 119910119891xyy) + 2119866119863119895

119899(minus119910119891xyy)

(9)

where



(10)




120590119909119909 = 119899sum119895=1

119895120590119909119909120590119910119910 = 119899sum

119895=1

119895120590119910119910120590119909119910 = 119899sum

119895=1

119895120590119909119910

(12)


120590119894119895 (119909 119910 119904)= minus119876120588119891119888119891 int119871119891

0119904120590119894119895119888119904 (119877 119904) 120597 (1199091015840 0 119904)

1205971199091015840 1198891199091015840 (13)

where



120585 = 119877radic 120588119898119888119898119904119870119898119890119891119891

(14)




(15)



Δ119879 (119909 119910 119905) 119889119910Δ119879 (119909 119910 119905) = 119879 (119909 119910 119905) minus 119879infin

(16)


120597119906119910120597119905 = 12 120597119908 (119909 119905)120597119905= (1 + V) 120572119879119870119898119890119891119891(1 minus V) 120588119898119888119898

120597Δ119879 (119909 119910 119905)120597119910100381610038161003816100381610038161003816100381610038161003816119910=0

(17)


1198791198980 minus 119879 (119909 119910 119905)1198791198980 minus 119879119894119899= 119890119891119888[[


1199102]]](18)


120597119879 (119909 119910 119905)120597119910100381610038161003816100381610038161003816100381610038161003816119910=0 =

2Δ1198791198602radic120587 exp [minus (1198601) 2] (19)



(20)

where A1 and A2 are

1198601 = 119909radic119870119898119890119891119891120588119898119888119898119876120588119891119888119891radic119905

1198602 = radic1205881198981198881198982radic119870119898119890119891119891119905

(21)












119879119891 (119909 0 0) = 119879119898 (119909 119910 0) = 1198790119908 (119909 0 0) = 1199080119875119891 (119909 0 0) = 119875119900119906119905

(24)

119879119891 (0 0 119905) = 119879119894119899119902 (0 0 119905) = 119902119894119899 (25)


1

3

4

N

5

2

Matrix

Interface

Central fracture



12059711987911989812059711991010038161003816100381610038161003816100381610038161003816(119909119871119898119905) = 0

12059711987911989812059711990910038161003816100381610038161003816100381610038161003816(0119905) =

12059711987911989812059711990910038161003816100381610038161003816100381610038161003816(119871119891119905) = 0

(26)












Matrix




coupling

IterationNo



stressYes


Nexttime

t+t


0

02

04

06

0

1

2

3

Dila

tion

(mm

)

Shea

r stre

ss (M

Pa)



n




00

02

04

06

08

10

0 50 100 150 200 250 300

Relat

ive t

empe

ratu

re in

frac

ture

Fracture length (m)

1 year

5 year

15 year



0098

0100

0102

0104

0106

0108

0110

0112

Ther

mal

indu

ced

half

aper

ture

(m)













0

04

08

12

16

2

0

100

200

300

400

500

600

700

800

900

Surf

ace p

ress

ure (

Psi)

Time


Inje

ctio

n ra

te (m

m

in)

2015

46

2013

56

2013

81

4

2013

11

22

2014

32

2014

61

0

2014

91

8

2014

12

27


0000

0005

0010

0015

0020

0025

0030

0035

Tota

l ape

rtur

e (m

)









0001

0010

0100

1000

10000

Aper

ture

(mm

)


0

30

60

90

120

150



Tem

pera

ture

(∘

)










Inlet


minus5

0

5

10

15

20

25

30

35

Def

orm

atio

n In

duce

d St

ress

(MPa

)



minus5

15

35

55

75

95

115D

efor

mat

ion

Indu

ced

Stre

ss (M

Pa)


Inlet




5 Conclusions









Inlet

minus35

minus30

minus25

minus20

minus15

minus10

minus05

00

Ther

mal

indu

ced

stres

s (M

Pa)




Inlet

minus50

minus40

minus30

minus20

minus10

00 Th

erm

al in

duce

d str

ess (

MPa

)




20

40

60

80

100

Max

imum

Hor

izon

tal S

tres

s (M

Pa)


05

1

15


00 05 1 15 2

Fracture Length (m)

Dist

ance

Alo

ng th

e Mat

rix

(m)





Data Availability




Acknowledgments


References






































Journal of












Journal of







Volume 2018






Volume 2018









y

x

(x1 y1)

(x y)2aj

(xjyj)

(xjyj)

(xn yn)

yx

j




119895120590119909119909 = 2119866119863119895119904(2119891xy + 119910119891xyy) + 2119866119863119895

119899(119891yy + 119910119891yyy)

119895120590119910119910 = 2119866119863119895119904(minus119910119891xyy) + 2119866119863119895

119899(119891yy minus 119910119891yyy)

119895120590119909119910 = 2119866119863119895119904(119891yy + 119910119891xyy) + 2119866119863119895

119899(minus119910119891xyy)

(9)

where



(10)




120590119909119909 = 119899sum119895=1

119895120590119909119909120590119910119910 = 119899sum

119895=1

119895120590119910119910120590119909119910 = 119899sum

119895=1

119895120590119909119910

(12)


120590119894119895 (119909 119910 119904)= minus119876120588119891119888119891 int119871119891

0119904120590119894119895119888119904 (119877 119904) 120597 (1199091015840 0 119904)

1205971199091015840 1198891199091015840 (13)

where



120585 = 119877radic 120588119898119888119898119904119870119898119890119891119891

(14)




(15)



Δ119879 (119909 119910 119905) 119889119910Δ119879 (119909 119910 119905) = 119879 (119909 119910 119905) minus 119879infin

(16)


120597119906119910120597119905 = 12 120597119908 (119909 119905)120597119905= (1 + V) 120572119879119870119898119890119891119891(1 minus V) 120588119898119888119898

120597Δ119879 (119909 119910 119905)120597119910100381610038161003816100381610038161003816100381610038161003816119910=0

(17)


1198791198980 minus 119879 (119909 119910 119905)1198791198980 minus 119879119894119899= 119890119891119888[[


1199102]]](18)


120597119879 (119909 119910 119905)120597119910100381610038161003816100381610038161003816100381610038161003816119910=0 =

2Δ1198791198602radic120587 exp [minus (1198601) 2] (19)



(20)

where A1 and A2 are

1198601 = 119909radic119870119898119890119891119891120588119898119888119898119876120588119891119888119891radic119905

1198602 = radic1205881198981198881198982radic119870119898119890119891119891119905

(21)












119879119891 (119909 0 0) = 119879119898 (119909 119910 0) = 1198790119908 (119909 0 0) = 1199080119875119891 (119909 0 0) = 119875119900119906119905

(24)

119879119891 (0 0 119905) = 119879119894119899119902 (0 0 119905) = 119902119894119899 (25)


1

3

4

N

5

2

Matrix

Interface

Central fracture



12059711987911989812059711991010038161003816100381610038161003816100381610038161003816(119909119871119898119905) = 0

12059711987911989812059711990910038161003816100381610038161003816100381610038161003816(0119905) =

12059711987911989812059711990910038161003816100381610038161003816100381610038161003816(119871119891119905) = 0

(26)












Matrix




coupling

IterationNo



stressYes


Nexttime

t+t


0

02

04

06

0

1

2

3

Dila

tion

(mm

)

Shea

r stre

ss (M

Pa)



n




00

02

04

06

08

10

0 50 100 150 200 250 300

Relat

ive t

empe

ratu

re in

frac

ture

Fracture length (m)

1 year

5 year

15 year



0098

0100

0102

0104

0106

0108

0110

0112

Ther

mal

indu

ced

half

aper

ture

(m)













0

04

08

12

16

2

0

100

200

300

400

500

600

700

800

900

Surf

ace p

ress

ure (

Psi)

Time


Inje

ctio

n ra

te (m

m

in)

2015

46

2013

56

2013

81

4

2013

11

22

2014

32

2014

61

0

2014

91

8

2014

12

27


0000

0005

0010

0015

0020

0025

0030

0035

Tota

l ape

rtur

e (m

)









0001

0010

0100

1000

10000

Aper

ture

(mm

)


0

30

60

90

120

150



Tem

pera

ture

(∘

)










Inlet


minus5

0

5

10

15

20

25

30

35

Def

orm

atio

n In

duce

d St

ress

(MPa

)



minus5

15

35

55

75

95

115D

efor

mat

ion

Indu

ced

Stre

ss (M

Pa)


Inlet




5 Conclusions









Inlet

minus35

minus30

minus25

minus20

minus15

minus10

minus05

00

Ther

mal

indu

ced

stres

s (M

Pa)




Inlet

minus50

minus40

minus30

minus20

minus10

00 Th

erm

al in

duce

d str

ess (

MPa

)




20

40

60

80

100

Max

imum

Hor

izon

tal S

tres

s (M

Pa)


05

1

15


00 05 1 15 2

Fracture Length (m)

Dist

ance

Alo

ng th

e Mat

rix

(m)





Data Availability




Acknowledgments


References






































Journal of












Journal of







Volume 2018






Volume 2018










(15)



Δ119879 (119909 119910 119905) 119889119910Δ119879 (119909 119910 119905) = 119879 (119909 119910 119905) minus 119879infin

(16)


120597119906119910120597119905 = 12 120597119908 (119909 119905)120597119905= (1 + V) 120572119879119870119898119890119891119891(1 minus V) 120588119898119888119898

120597Δ119879 (119909 119910 119905)120597119910100381610038161003816100381610038161003816100381610038161003816119910=0

(17)


1198791198980 minus 119879 (119909 119910 119905)1198791198980 minus 119879119894119899= 119890119891119888[[


1199102]]](18)


120597119879 (119909 119910 119905)120597119910100381610038161003816100381610038161003816100381610038161003816119910=0 =

2Δ1198791198602radic120587 exp [minus (1198601) 2] (19)



(20)

where A1 and A2 are

1198601 = 119909radic119870119898119890119891119891120588119898119888119898119876120588119891119888119891radic119905

1198602 = radic1205881198981198881198982radic119870119898119890119891119891119905

(21)












119879119891 (119909 0 0) = 119879119898 (119909 119910 0) = 1198790119908 (119909 0 0) = 1199080119875119891 (119909 0 0) = 119875119900119906119905

(24)

119879119891 (0 0 119905) = 119879119894119899119902 (0 0 119905) = 119902119894119899 (25)


1

3

4

N

5

2

Matrix

Interface

Central fracture



12059711987911989812059711991010038161003816100381610038161003816100381610038161003816(119909119871119898119905) = 0

12059711987911989812059711990910038161003816100381610038161003816100381610038161003816(0119905) =

12059711987911989812059711990910038161003816100381610038161003816100381610038161003816(119871119891119905) = 0

(26)












Matrix




coupling

IterationNo



stressYes


Nexttime

t+t


0

02

04

06

0

1

2

3

Dila

tion

(mm

)

Shea

r stre

ss (M

Pa)



n




00

02

04

06

08

10

0 50 100 150 200 250 300

Relat

ive t

empe

ratu

re in

frac

ture

Fracture length (m)

1 year

5 year

15 year



0098

0100

0102

0104

0106

0108

0110

0112

Ther

mal

indu

ced

half

aper

ture

(m)













0

04

08

12

16

2

0

100

200

300

400

500

600

700

800

900

Surf

ace p

ress

ure (

Psi)

Time


Inje

ctio

n ra

te (m

m

in)

2015

46

2013

56

2013

81

4

2013

11

22

2014

32

2014

61

0

2014

91

8

2014

12

27


0000

0005

0010

0015

0020

0025

0030

0035

Tota

l ape

rtur

e (m

)









0001

0010

0100

1000

10000

Aper

ture

(mm

)


0

30

60

90

120

150



Tem

pera

ture

(∘

)










Inlet


minus5

0

5

10

15

20

25

30

35

Def

orm

atio

n In

duce

d St

ress

(MPa

)



minus5

15

35

55

75

95

115D

efor

mat

ion

Indu

ced

Stre

ss (M

Pa)


Inlet




5 Conclusions









Inlet

minus35

minus30

minus25

minus20

minus15

minus10

minus05

00

Ther

mal

indu

ced

stres

s (M

Pa)




Inlet

minus50

minus40

minus30

minus20

minus10

00 Th

erm

al in

duce

d str

ess (

MPa

)




20

40

60

80

100

Max

imum

Hor

izon

tal S

tres

s (M

Pa)


05

1

15


00 05 1 15 2

Fracture Length (m)

Dist

ance

Alo

ng th

e Mat

rix

(m)





Data Availability




Acknowledgments


References






































Journal of












Journal of







Volume 2018






Volume 2018









1

3

4

N

5

2

Matrix

Interface

Central fracture



12059711987911989812059711991010038161003816100381610038161003816100381610038161003816(119909119871119898119905) = 0

12059711987911989812059711990910038161003816100381610038161003816100381610038161003816(0119905) =

12059711987911989812059711990910038161003816100381610038161003816100381610038161003816(119871119891119905) = 0

(26)












Matrix




coupling

IterationNo



stressYes


Nexttime

t+t


0

02

04

06

0

1

2

3

Dila

tion

(mm

)

Shea

r stre

ss (M

Pa)



n




00

02

04

06

08

10

0 50 100 150 200 250 300

Relat

ive t

empe

ratu

re in

frac

ture

Fracture length (m)

1 year

5 year

15 year



0098

0100

0102

0104

0106

0108

0110

0112

Ther

mal

indu

ced

half

aper

ture

(m)













0

04

08

12

16

2

0

100

200

300

400

500

600

700

800

900

Surf

ace p

ress

ure (

Psi)

Time


Inje

ctio

n ra

te (m

m

in)

2015

46

2013

56

2013

81

4

2013

11

22

2014

32

2014

61

0

2014

91

8

2014

12

27


0000

0005

0010

0015

0020

0025

0030

0035

Tota

l ape

rtur

e (m

)









0001

0010

0100

1000

10000

Aper

ture

(mm

)


0

30

60

90

120

150



Tem

pera

ture

(∘

)










Inlet


minus5

0

5

10

15

20

25

30

35

Def

orm

atio

n In

duce

d St

ress

(MPa

)



minus5

15

35

55

75

95

115D

efor

mat

ion

Indu

ced

Stre

ss (M

Pa)


Inlet




5 Conclusions









Inlet

minus35

minus30

minus25

minus20

minus15

minus10

minus05

00

Ther

mal

indu

ced

stres

s (M

Pa)




Inlet

minus50

minus40

minus30

minus20

minus10

00 Th

erm

al in

duce

d str

ess (

MPa

)




20

40

60

80

100

Max

imum

Hor

izon

tal S

tres

s (M

Pa)


05

1

15


00 05 1 15 2

Fracture Length (m)

Dist

ance

Alo

ng th

e Mat

rix

(m)





Data Availability




Acknowledgments


References






































Journal of












Journal of







Volume 2018






Volume 2018












Matrix




coupling

IterationNo



stressYes


Nexttime

t+t


0

02

04

06

0

1

2

3

Dila

tion

(mm

)

Shea

r stre

ss (M

Pa)



n




00

02

04

06

08

10

0 50 100 150 200 250 300

Relat

ive t

empe

ratu

re in

frac

ture

Fracture length (m)

1 year

5 year

15 year



0098

0100

0102

0104

0106

0108

0110

0112

Ther

mal

indu

ced

half

aper

ture

(m)













0

04

08

12

16

2

0

100

200

300

400

500

600

700

800

900

Surf

ace p

ress

ure (

Psi)

Time


Inje

ctio

n ra

te (m

m

in)

2015

46

2013

56

2013

81

4

2013

11

22

2014

32

2014

61

0

2014

91

8

2014

12

27


0000

0005

0010

0015

0020

0025

0030

0035

Tota

l ape

rtur

e (m

)









0001

0010

0100

1000

10000

Aper

ture

(mm

)


0

30

60

90

120

150



Tem

pera

ture

(∘

)










Inlet


minus5

0

5

10

15

20

25

30

35

Def

orm

atio

n In

duce

d St

ress

(MPa

)



minus5

15

35

55

75

95

115D

efor

mat

ion

Indu

ced

Stre

ss (M

Pa)


Inlet




5 Conclusions









Inlet

minus35

minus30

minus25

minus20

minus15

minus10

minus05

00

Ther

mal

indu

ced

stres

s (M

Pa)




Inlet

minus50

minus40

minus30

minus20

minus10

00 Th

erm

al in

duce

d str

ess (

MPa

)




20

40

60

80

100

Max

imum

Hor

izon

tal S

tres

s (M

Pa)


05

1

15


00 05 1 15 2

Fracture Length (m)

Dist

ance

Alo

ng th

e Mat

rix

(m)





Data Availability




Acknowledgments


References






































Journal of












Journal of







Volume 2018






Volume 2018









00

02

04

06

08

10

0 50 100 150 200 250 300

Relat

ive t

empe

ratu

re in

frac

ture

Fracture length (m)

1 year

5 year

15 year



0098

0100

0102

0104

0106

0108

0110

0112

Ther

mal

indu

ced

half

aper

ture

(m)













0

04

08

12

16

2

0

100

200

300

400

500

600

700

800

900

Surf

ace p

ress

ure (

Psi)

Time


Inje

ctio

n ra

te (m

m

in)

2015

46

2013

56

2013

81

4

2013

11

22

2014

32

2014

61

0

2014

91

8

2014

12

27


0000

0005

0010

0015

0020

0025

0030

0035

Tota

l ape

rtur

e (m

)









0001

0010

0100

1000

10000

Aper

ture

(mm

)


0

30

60

90

120

150



Tem

pera

ture

(∘

)










Inlet


minus5

0

5

10

15

20

25

30

35

Def

orm

atio

n In

duce

d St

ress

(MPa

)



minus5

15

35

55

75

95

115D

efor

mat

ion

Indu

ced

Stre

ss (M

Pa)


Inlet




5 Conclusions









Inlet

minus35

minus30

minus25

minus20

minus15

minus10

minus05

00

Ther

mal

indu

ced

stres

s (M

Pa)




Inlet

minus50

minus40

minus30

minus20

minus10

00 Th

erm

al in

duce

d str

ess (

MPa

)




20

40

60

80

100

Max

imum

Hor

izon

tal S

tres

s (M

Pa)


05

1

15


00 05 1 15 2

Fracture Length (m)

Dist

ance

Alo

ng th

e Mat

rix

(m)





Data Availability




Acknowledgments


References






































Journal of












Journal of







Volume 2018






Volume 2018













0

04

08

12

16

2

0

100

200

300

400

500

600

700

800

900

Surf

ace p

ress

ure (

Psi)

Time


Inje

ctio

n ra

te (m

m

in)

2015

46

2013

56

2013

81

4

2013

11

22

2014

32

2014

61

0

2014

91

8

2014

12

27


0000

0005

0010

0015

0020

0025

0030

0035

Tota

l ape

rtur

e (m

)









0001

0010

0100

1000

10000

Aper

ture

(mm

)


0

30

60

90

120

150



Tem

pera

ture

(∘

)










Inlet


minus5

0

5

10

15

20

25

30

35

Def

orm

atio

n In

duce

d St

ress

(MPa

)



minus5

15

35

55

75

95

115D

efor

mat

ion

Indu

ced

Stre

ss (M

Pa)


Inlet




5 Conclusions









Inlet

minus35

minus30

minus25

minus20

minus15

minus10

minus05

00

Ther

mal

indu

ced

stres

s (M

Pa)




Inlet

minus50

minus40

minus30

minus20

minus10

00 Th

erm

al in

duce

d str

ess (

MPa

)




20

40

60

80

100

Max

imum

Hor

izon

tal S

tres

s (M

Pa)


05

1

15


00 05 1 15 2

Fracture Length (m)

Dist

ance

Alo

ng th

e Mat

rix

(m)





Data Availability




Acknowledgments


References






































Journal of












Journal of







Volume 2018






Volume 2018











0001

0010

0100

1000

10000

Aper

ture

(mm

)


0

30

60

90

120

150



Tem

pera

ture

(∘

)










Inlet


minus5

0

5

10

15

20

25

30

35

Def

orm

atio

n In

duce

d St

ress

(MPa

)



minus5

15

35

55

75

95

115D

efor

mat

ion

Indu

ced

Stre

ss (M

Pa)


Inlet




5 Conclusions









Inlet

minus35

minus30

minus25

minus20

minus15

minus10

minus05

00

Ther

mal

indu

ced

stres

s (M

Pa)




Inlet

minus50

minus40

minus30

minus20

minus10

00 Th

erm

al in

duce

d str

ess (

MPa

)




20

40

60

80

100

Max

imum

Hor

izon

tal S

tres

s (M

Pa)


05

1

15


00 05 1 15 2

Fracture Length (m)

Dist

ance

Alo

ng th

e Mat

rix

(m)





Data Availability




Acknowledgments


References






































Journal of












Journal of







Volume 2018






Volume 2018










Inlet


minus5

0

5

10

15

20

25

30

35

Def

orm

atio

n In

duce

d St

ress

(MPa

)



minus5

15

35

55

75

95

115D

efor

mat

ion

Indu

ced

Stre

ss (M

Pa)


Inlet




5 Conclusions









Inlet

minus35

minus30

minus25

minus20

minus15

minus10

minus05

00

Ther

mal

indu

ced

stres

s (M

Pa)




Inlet

minus50

minus40

minus30

minus20

minus10

00 Th

erm

al in

duce

d str

ess (

MPa

)




20

40

60

80

100

Max

imum

Hor

izon

tal S

tres

s (M

Pa)


05

1

15


00 05 1 15 2

Fracture Length (m)

Dist

ance

Alo

ng th

e Mat

rix

(m)





Data Availability




Acknowledgments


References






































Journal of












Journal of







Volume 2018






Volume 2018











Inlet

minus35

minus30

minus25

minus20

minus15

minus10

minus05

00

Ther

mal

indu

ced

stres

s (M

Pa)




Inlet

minus50

minus40

minus30

minus20

minus10

00 Th

erm

al in

duce

d str

ess (

MPa

)




20

40

60

80

100

Max

imum

Hor

izon

tal S

tres

s (M

Pa)


05

1

15


00 05 1 15 2

Fracture Length (m)

Dist

ance

Alo

ng th

e Mat

rix

(m)





Data Availability




Acknowledgments


References






































Journal of












Journal of







Volume 2018






Volume 2018










Data Availability




Acknowledgments


References






































Journal of












Journal of







Volume 2018






Volume 2018



















Journal of












Journal of







Volume 2018






Volume 2018








coupled thm and matrix stability modeling of hydroshearing

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