Research ArticleFinite Element Optimised Back Analysis ofIn Situ Stress Field and Stability Analysis of Shaft Wall inthe Underground Gas Storage

Yifei Yan1 Bing Shao1 Jianguo Xu2 and Xiangzhen Yan2

1College of Mechanical and Electronic Engineering China University of Petroleum Qingdao 266580 China2College of Storage and Transportation and Architectural Engineering China University of Petroleum Qingdao 266580 China

Correspondence should be addressed to Yifei Yan yanyf163163com

Received 14 August 2015 Revised 11 January 2016 Accepted 13 January 2016

Academic Editor Masoud Hajarian

Copyright copy 2016 Yifei Yan et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A novel optimised back analysis method is proposed in this paper The in situ stress field of an underground gas storage (UGS)reservoir in a Turkey salt cavern is analysed by the basic theory of elastic mechanics A finite element method is implemented tooptimise and approximate the objective function by systematically adjusting boundary loads Optimising calculation is performedbased on a novel method to reduce the error between measurement and calculation as much as possible Compared with commonback analysis methods such as regression method the method proposed can further improve the calculation precision Byconstructing a large circular geometric model the effect of stress concentration is eliminated and a minimum difference betweencomputed and measured stress can be guaranteed in the rectangular objective region The efficiency of the proposed method isinvestigated and confirmed by its capability on restoring in situ stress field which agrees well with experimental results Thecharacteristics of stress distribution of chosen UGS wells are obtained based on the back analysis results and by applying thecorresponding fracture criterion the shaft walls are proven safe

1 Introduction

In situ stress is an important factor that should be consideredduring the construction of underground gas storage (UGS)A direct way to get the in situ stress is tomeasure the principalstress by test methods such as micro-fracturing test [1]

As the engineering technology develops kinds of tech-niques have been selectively accepted for obtaining accuratein situ stress orientation and magnitude Hydraulic frac-turing when poroelastic effects are considered seems tobe the best option to derive the horizontal in situ stressmagnitude and direction [2] If a vertical fracture is createdin the process then the minimum in situ stress must be inthe horizontal direction Although fracturing has been themost commonly used method in in situ stress measuringit may not be available or even fail sometimes for examplein inclined test holes where the fracture direction is notnormal to the minimum horizontal stress On the otherhand the fracturing method will also be ineffective when

themeasurement requires a higher injection pressure and theterrain with UGS located is bearing a very large stress Someresearchers implement methods such as leak-off tests andbreakout orientation to the standard well testing and logginghowever these methods are not always reliable Moreoverthe stresses obtained from breakout analysis are confusing asbreakout can be induced by different kinds of in situ stresses

When in situ stress data from test wells is obtainedsome techniques can be used to identify the magnitude anddirection of the whole in situ stress field [3]These techniquesare organised as an overdetermined system of equationsand the in situ stress state can be solved by using severalmethods mentioned in some articles [4ndash6] Not only can themagnitude of the two principal stresses be solved but alsothe direction of the field can be determined However themethods assume that the shear stress is negligible as it is zerowhen the normal stress is a principal stress Djurhuus andAadnoslashy extended the study on in situ stress and presenteda new theory [4] They obtained experiment data from

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 4052483 12 pageshttpdxdoiorg10115520164052483

2 Mathematical Problems in Engineering

multiple fracturing and induced fractures from imaginglogs to calculate the in situ stress state The magnitudeinformation is mainly calculated from fracturing data whilethe orientation information can be obtained from image logs

However measurement is difficult to carry out becauseof high cost on an injection-production well Thereforeseveral in situ stress values can only be obtained throughexperiments The stress distribution of the test zone can becalculated quicklywith the help of finite element optimisationand the results can be more accurate with some optimi-sation methods implemented To carry out the inversioncalculation we need to construct the finite element modelwith proper boundary conditions and loads Some minormodifications should also be incorporated into the modelto eliminate the effect of stress concentration which will bediscussed below

Boundary conditions and loading methods cannot beinput directly because the tectonic movement is unknownand the geologic structure is complicated These circum-stances make in situ stress field analysis difficult to carryout So boundary load inversion comes to be the crucialway to conduct research on the in situ stress field Fromthe viewpoint of geomechanics the tectonic stress field isthe in situ stress field forming the tectonic system and typeincluding its distribution area and inside stress distributionDuring the analysis rock density could be determined rel-atively accurately in a small variation range with differentprecision levels of gravity stress and tectonic stress Instead ofbeing inversed based on terrain conditions the finite elementmethod could be used to calculate gravity stress to ensure theprecision can satisfy the engineering requirement Howevertectonic stress field calculation is different [7] During theconstruction of UGS the distribution of the in situ stress fieldalong the plane at the top of the salt cavern is needfulwhile thein situ stress information of only a fraction ofwells in a certainarea is availableThen inversion and extrapolation need to beperformed to obtain the stress field of the whole area Amongthe present numerical simulations and analysis methods ofthe in situ stress field the most widely used approach iscut-and-try and adjustment of boundary loads [8ndash13] Giventhe complexity of the geological structure and the limitedreference of the in situ stress boundary conditions andloading methods become the difficult problem in calculationand analysis part of a numerical simulation The major issueis how to improve the precision of the inversion and reliabilityof the data Simulation is an effective way to take terrain andgeological features into consideration And only by this canrelevant theoretical or numerical analyses such as back anal-ysis back calculation and simulation be performed to predictthe tectonic stress and its mode based on the measureddata [12]The boundary loads identification and optimisationmethod presented by this paper adopts damped least squaresalgorithm to approximate the loads by setting parametersfrom stress testing references The initial value for iterationis got by the unit load method to avoid the limits from thestructure geometric features and boundary conditions Bycontrolling the error of observing point the identification onuncertain and unknown boundary loads of the in situ stressfield is achieved A few ways are attempted and compared to

improve the data reliability and the precision of in situ stressfield inversion calculation in this paper

2 Building the Finite Element Model

It is difficult to achieve an analytical solution of boundaryloads for a formation due to the complexity of the geolog-ical structure and boundary conditions The finite elementmethod is employed here for mechanical analysis to makesure that the research method can have a general application

Though salt is viscoelastic aUGS reservoir stress field canbe treated as an elastic problem to simplify the analysis Thisstudy focuses on the original rock salt without consideringthe effect of creep and relax The linear elastic assumptionmay bring some deviation with the actual situation but it ishigh needed for figuring out a specific engineering problem

The applied loads generate displacement field stress fieldand strain field inside for a certain area The basic equationsof small deformation elastic mechanics are as follows

equilibrium differential equation

119871119879120590 + 119876 = 0 (1)

geometric equation

120576 = 119871119880 (2)

physical equation

120590 = 119863120576 (3)

where 119871 is the differential operator and119863 is the elastic matrixdepending on the elastic modulus 119864 of the formation andPoissonrsquos ratio ]

Generally the finite element formula is

119870119880 = 119865 (119883) (4)

where119870 is the global stiffness matrixU is the node displace-ment matrix 119865(119883) is the boundary load array and 119883 is theparameter vector used to describe the boundary loads of UGSreservoir

Equation (4) is transformed to

119880 = 119870minus1119865 (119883) (5)

Equation (2) is substituted in (3) to get the followingequation

120590 = 119863119871119880 (6)

Then (5) is substituted in (6) that is

120590 = 119863119871119870minus1119865 (119883) (7)

From (7) the nodal stress and the boundary loads satisfythe following equation

120590 = 119879119865 (119883) (8)

Mathematical Problems in Engineering 3

where119879 = 119863119871119870minus1119879 is the transfermatrix between the nodal

stress matrix and the boundary loads array For the linearelastic body 119879 could be determined when the finite elementmodel of UGS reservoir is established 119865(119883) is uncertainbecause the loads are unknown and only a part of120590 is certainsince the stress values of only a few wells in the reservoir tobe studied are known Therefore in general the boundaryloads parameter vector 119883 cannot be calculated from (8) andis waiting to be discovered

3 Optimisation Method

In situ stress field inversion is used to find the model orparameters that can fit actual data under certain principlesIt is not only a method of data fitting but also a way ofoptimising The inversion result is closely related to theoptimal criterions which is used to evaluate the degree ofdata fitting Sometimes we have prior information aboutsome areas that have already been researched Based on thisprior information we could restrain the optimal criterion andparameters Generally the data going to be used in stressinversion calculation is obtained from the reservoir stressfield and contains the stress of logging fracturing boreholebreakout or core testing of certain wells The principle ofstress inversion (or objective function) could be described asthe followingmodels by using the least squaresmethod as theoptimal criterion

(1) Constraint Model If the stress value of the test well is 120590119871

that is the simulation value of the stress field 120590 from the finiteelement calculation is known the model could be describedas

min Φ (119883) = (120590 minus 120590119871)119879(120590 minus 120590

119871)

st ℎ1le 120593 (119883) le ℎ

2

119886119894le 119883119894le 119887119894

(9)

where 120590 is derived from the finite element calculation Φ(119883)

is the mean squared error between the measurement andcalculation 119883 is the unknown boundary load vector andℎ1and ℎ

2are the boundary load parameter functions of the

upper bound and lower bound respectively determined byprior information 119886

119894and 119887119894are the boundary load parameters

of upper and lower bound respectively

(2) Unconstraint Model An unconstraint model exists whenthe condition in (9) is not met that is

minΦ (119883) = (120590 minus 120590119871)119879(120590 minus 120590

119871) (10)

where symbols are the same as those in (9)

4 Inversion Algorithm

According to the principle of inversion calculation backanalysis is used to calculate the input data of a system byanalysing its output data [13] Therefore if the system is wellconstructed it is possible to obtain the input data from its

output data which is usually achieved through experimentsThe back analysis ensures that a unique solution can begenerated as long as the mechanical model is real and thedata is sufficient

The general way of inversion is to present the parametervector of boundary load 119883 and then calculate the reservoirrsquosstress response value under 119883 During this process thefinite element method is used to approach the actual data ofsome well sites based on the least squares method Usuallythe Gauss-Newton method is adopted to solve the leastsquares problem of (9) or (10) However ill-conditionedcoefficient matrix of the equation low iterative speed andspurious convergence may occur sometimes In this paper adamped least squares method [Levenberg-Marquardt (LM)method] [14] is implemented to approach the parametervector of boundary load 119883 The LM method which has ahigh optimisation speed and precision possesses both theadvantages of the gradient method and the Newton methodWhen the initial value of damping factor 120582

119896is small the step

size is set to be equal to that of the Newton method while itis equal to that of the gradient descent when 120582

119896is large

When the iteration reaches step k the column vector ofstress response and boundary load120590

119896 and119883119896 are setThen120590119896

series expansion around119883119896 is performed and the trace above

the second-order polynomial is ignored to get the followingequation [15]

120590119896= 120590119896minus1

+120597120590

120597119883(119883119896minus 119883119896minus1

)

= 120590119896minus1

+ 119869 sdot (119883119896minus 119883119896minus1

)

(11)

where 119869 is the Jacobi matrix of 120590119896 in119883119896 and it is defined from

the finite difference method as

119869119894119895=

120597120590119894

120597119883119895

asympΔ120590119894

Δ119883119895

=

120590119894(1198831 1198832 119883

119895+ Δ119883119895 119883

119899)

Δ119883119895

minus

120590119894(1198831 1198832 119883

119895 119883

119899)

Δ119883119895

(12)

where 120590119894could be calculated from (7)

The search direction used in LM method is from thesolution of a set of linear equalities The iterative formula is

[119869119879(119883(119896)

) 119869 (119883(119896)

) + 120582119896119868] Δ119883

(119896)

= minus119869119879(119883(119896)

) sdot 120590119896minus1

minus 120590119871

(13)

where Δ119883 is the search direction vector of 119883 and 119868 is theunit matrix Scalar 120582

119896is the damping factor controlling the

direction and value ofΔ119909119896When 120582

119896is 0 the direction ofΔ119909

119896

is the same as the result of the Gauss-Newton method When

4 Mathematical Problems in Engineering

120582119896tends to infinity we could derive the following equation

from (13)

[119869119879(119883(119896)

) 119869 (119883(119896)

) + 120582119896119868] Δ119883

(119896) asymp 120582119896119868 Δ119883

(119896)

= minus119869119879(119883(119896)

) sdot 120590119896minus1

minus 120590119871

(14)

Thus

Δ119883(119896)

= minus1

120582119896

119869119879(119883(119896)

) sdot 120590119896minus1

minus 120590119871 (15)

where Δ119883(119896) is in the direction of the zero vector and has a

steep descending slope for 120582119896 which is large enough that is

Φ(119909(119896)

+ Δ119909(119896)

) lt Φ (119909(119896)

) (16)

From (13) we derived

Δ119883(119896)

= minus [119869119879(119883(119896)

) 119869 (119883(119896)

) + 120582119896119868]minus1

119869119879(119883(119896)

)

sdot 120590119896minus1

minus 120590119871

(17)

Then we got

119883(119896)

= 119883(119896minus1)

+ Δ119883(119896)

(18)

The boundary load vector could be derived from iterativesolving of (13)ndash(18) The key step is how to choose iterativeinitial value If the range of the boundary load is knownwe can set the iterative initial value of 119883 as the average ofthe upper and lower bounds If the range is an unboundeddomain the iterative initial value of 119883 could be determinedafter experiential trials

From the above discussion if the measured data ofreservoir stress is available the boundary load could be deter-mined The load parameter vector and reservoirrsquos horizontalstress 120590

119871could satisfy

119866119883 = 120590119871 (19)

where 119866 isin 119862119898times119899 denotes the transfer matrix 119898 is the point

number for the measurement and 119899 is the dimension of theboundary load vector If 119899 gt 119898 a set of boundary loadvectors could be clear based on the principle of controlling thecomprehensive error to be minimum for the nonuniquenessof the solution

The specific steps of finite element inversion of the in situstress field go as follows Firstly choose a proper objectivefunction Secondly adjust and search parameters by theoptimal method and forward-model the computed value bythe finite element numerical algorithm Thirdly substitutethe measured and calculated values into the objective func-tion and judge the result from the objective function Ifthe result did not reach the minimum the adjusting andsearching should be continued Repeat this process untilthe result from objective function reaches the minimumvalue and finally the boundary load parameter is generatedAfterwards the forward modelling is carried out to acquirethe reservoirrsquos stress field by the boundary load derived

Initial data X

Parameter modification

Inversion parameter and results

Feature analysis of in situ stress field

Forward numerical calculation 120590 = TF(X)

Objective function 120601(X)

Measured data 120590L

120601(X) le 120576

Figure 1 Flowchart of the finite element inversion

Object

P1

P2

P3

P4

P5

P6

P7

P8

P9

P10

P11

P12

Figure 2 Computational model for the boundary load

from the inversion (Figure 1 shows the inversion process)The applied boundary condition is shown in Figure 2 Thecalculation is performed in the entire circle area to avoidstress concentration Assuming the strata are boundless andstress is continuously changed if the stress applied on thecircle can make the inversion results close enough to the testresults the stress applied on the rectangular must be rightThe stress distribution in the object area is the result we wantfrom the inversion

5 Example and Application

Based on the aforementioned theory in order to get theboundary load of the in situ stress field we could programthe finite element iteration by finite element software Inthis paper a Turkey salt cavern is chosen to illustrate theapplication of reversion technology on resolving in situ stressboundary load Furtherly this method is also applied on thein situ stress analysis in Qinshui basin of China [16ndash18] Theleast moving squares method can also be used to deduce

Mathematical Problems in Engineering 5

1150 1100 1150 1150 1200 1200 1250 125012501100

1250

Tuz formation

Cihanbeyli to Kochisar formation

1100

1800

1600

1400

1200

1000

800

600

400

200

0

Dep

th b

elow

surfa

ce (m

)

UG

S12

UG

S11

UG

S10

UG

S1

UG

S2

UG

S9

UG

S8

UG

S7

UG

S6

UG

S5

UG

S4

UG

S3

Figure 3 Section map of UGS in Turkey salt cavern

UGS11

UGS10

UGS1

UGS2UGS9

UGS8

UGS7

UGS6

UGS5 UGS4UGS3

Figure 4 Three-dimensional overview of UGS in Turkey salt cavern

the in situ stress field by carefully selecting the primaryfunction and the weight function [19] The UGS distributionis shown in Figures 3ndash5 Figure 3 shows the well depths ofmultiple UGS wells The minimum depth is 1110m whilethe maximum depth is 1500m Figure 4 shows the three-dimensional distribution of these UGS wells Figure 5 showsthe planar structure of the salt cavern Twelve reservoirs asmany as UGS wells construct the salt cavern All sites are inthe southeast section which is indicated by the blue spotsThe red line in Figure 5 indicates the fault zone of the researcharea

51 Building UGS in a Salt Cavern Salt owns several proper-ties that make it ideal for gas storage [19] It has a moderatelyhigh strength and flows plastically to close fracture that couldotherwise become a leak The porosity and permeability ofsalt to liquid and gaseous hydrocarbons are close to zerothereby preventing stored gas escape [20 21] Salt caverns canprovide high deliverability and gas can be withdrawn quicklybecause no pressure from flow is lost through the pores Asalt cavern in south Turkey is used to construct undergroundnatural gas storage to compensate season-related fluctuations

of gas consumption Moreover by a salt cavern gas can beprovided to customers under optimum economic conditionsbetween buying and selling To construct UGS by a saltcavern a well should be drilled first and then water shouldbe injected to dissolve the salt Brine is pumped out of the saltcavern and a cavity is formed which can be treated to storenatural gas

52 Calculation Model and Boundary Conditions Based onthe prerequisite of the numerical simulation a geologicmodel the target layer isolation body of a rock block is takenas the simulation object Two rules should be followed todetermine the computational area (1) extend the geometricrange to eliminate or weaken the effects from the bound-ary conditions on the research area and (2) the geometryconstraint conditions on the boundary are easy to get Ingeneral the layered distribution of underground sedimentaryrock could be treated as a plane problem of elastic mechanicsin the comprehensive mathematical model Figure 5 is themechanical model of the research area which is a rectangulararea with a length of 8 km and width of 32 km Three majorfault zones are taken into consideration The lengths and

6 Mathematical Problems in Engineering

UGS11UGS12

UGS1UGS10

UGS2 UGS9UGS8 UGS7

UGS6 UGS5UGS4 UGS3

0 1000 2000(m)

Figure 5 Planar structure of the research area and the distributionof the UGS

widths of fault zones are 8 km lowast 1 km 12 km lowast 08 km and35 km lowast 08 km respectively

The following eight factors are taken as the basic factorsof tectonic force compressive or tensile forces and shearforces in the east west south and north The boundaryload is adjusted depending on the size of research area Asthe stress field is continuously changed the external stressin the circle can be divided into 12 parts which is largeenough to guarantee the accuracy of calculation By changingthe value of the 12 boundary forces we could simulate thecompressive or tensile forces in the direction of eastndashwest andsouthndashnorth as well as shear deformation in the horizontalplane to get the in situ stress field distribution of the partialresearch area At the same time boundary force constraintis applied to make the value of the resultant force and theresultant moment zero (Figure 6 shows the method to applythe boundary force)The edge points of the lower boundary ofthemodel are fixed to ensure no rigid translation and rotationin the research area

Generally a geologic body is taken as a rock with auniform block to confirm the rock physical parameters fromdifferent tectonic positions The basic assumption is thateach fault zone is homogeneous The rock strength Youngrsquosmodulus and other rock mechanical parameters should bedecreased according to the rock properties in specific depth(Table 1 shows information about mechanical parameterschosen by the model) The cored sample is taken from the

1ElementsMAT Num

Figure 6 Mechanical model of the calculated area

Table 1 Rock mechanics parameters

Number Media type Elasticity modulusMPa Poissonrsquos ratio ]1 Reservoir 31200 02252 Fault zone 1 22500 02703 Fault zone 2 23500 02604 Fault zone 3 24000 02905 Other layers 35200 0220

spot field and by indoor experiment elasticity modulus andPoissonrsquos ratio are obtained The grids of the model adoptedmainly are four-node isoperimetric rectangular elements andfor part of the area triangle elements are used In additionthe element length of the fault zone and its surrounding areais shorter and the grids are relatively finer than those of otherparts

53 Inversion of the In Situ Stress Field of the Salt Cavern

(1) Initial boundary conditions the initial values of thedistributed normal stress 119875

1sim 11987512in the east south

west and north could be set as an arbitrary value with10MPa here (Figure 6 shows how the boundary forceacts)

(2) Constraint object based on the reference about insitu stress measurement we chose 10 values of in situstress and five directions from eight survey points asknown constraint parameters to inverse 12 unknownboundary forces (Figure 5 shows the distribution ofconstraint points and measured data are shown inTables 2 and 3)

(3) Searching range the range of the boundary nor-mal stress constraint value is 1MPasim30MPa (lowerboundsimupper bound) and in the same way therange of the boundary shear stress constraint valueminus10MPasim10MPa The precision of the iteration ter-mination is 10minus5MPa2The boundary force constraintshould be taken into consideration to make sure thatthe resultant moment of the boundary force is zero

Mathematical Problems in Engineering 7

Table 2 Comparison between the inversion results and the measured values on principal stress magnitude

Measured points Stress components Inversion valueMPa Measured valueMPa Relative error

UGS6 120590119867

minus2169 minus2323 66120590ℎ

minus2793 minus2777 058

UGS7 120590119867

minus2132 minus1996 681120590ℎ

minus2602 minus261 031

UGS8 120590119867

minus2121 minus2026 469120590ℎ

minus2725 minus2852 445

UGS9 120590119867

minus2066 minus2226 719120590ℎ

minus2783 minus2664 447

UGS10 120590119867

minus1906 minus1856 269120590ℎ

minus2515 minus2385 545

Table 3 Comparison between the inverted results and themeasuredvalues on the direction of maximum principal stress

Measuredpoints Inversion value(∘) Measured value(∘) Error(∘)

UGS6 NE57327 NE55 2327UGS7 NE51798 NE535 minus1702

UGS8 NE52334 NE549 minus257

UGS9 NE4132 NE451 minus378

UGS10 NE51713 NE536 minus1887

The relevant constrained inversion model could bederived from (9)

(4) Inversion result based on the conditions above thecalculation results on the survey points t from inver-sion are shown in Tables 2 and 3 where 120590

119867and

120590ℎare the maximum and minimum principal stress

magnitude respectively The measurement resultsfrom well logging are also presented and comparedwith the inversion results Table 4 shows the results ofthe boundary load inversion

Tables 2 and 3 show that the maximum relative errorsof stress components 120590

119867and 120590

ℎare controlled within 8

The errors of the direction of maximum principal stress arecontrolled within 4∘ All the stress values obtained from theinversion reached or are close to the actual data and theprecision of the inversion results could meet engineeringrequirements

We applied the optimised boundary force from the inver-sion to perform forward calculation This approach couldlead to the distribution of the stress field value and directionof the Turkey salt cavern (shown in Figures 7 and 8)

6 Stress Distribution Character of the Wellnear the Hole of the UGS

On the basis of the analysis above the two principal stresses120590119867and 120590ℎof the 12 wells are determined Given that the stress

concentration occurs by the hole which will influence thestability of the shaft wall the von Mises stress distributioncharacter near the shaft wall (0ndash255m) should be checked

minus29

minus13minus14

minus15

minus16

minus17

minus18

minus19

minus20

minus21

minus22

minus23

minus24

minus25

minus26

minus27

minus28

minus29

minus30

minus31

minus32

minus33

minus34

minus24

minus19

minus19

minus19

minus19

minus19minus19

2000

4000

6000

8000

10000

12000

14000

4000 6000 8000 100002000

Figure 7 Distribution of the maximum principal stress and orien-tation

up as shown in Figure 9 Here well UGS1 is taken as anexample with well diameter of UGS1 279 cm bottom holepressure 12MPa overburden pressure 231MPa and welldepth 1100m

As shown in Figure 9 given that stress nonhomogeneouswith the in situ stress exists the von Mises stress distributionof the shaft wall is in an oval shape The von Mises stresshas a higher value in the shaft wall which is within 281ndash568MPa The von Mises stress decreases rapidly away fromthe shaft wall and the shape changes to a butterfly-like oneThe maximum von Mises stress decreases to 171MPa when06m away from the shaftwall It is worthmentioning that theresult here contains the influence from stress concentrationaround the shaft wall As shown in Figures 9(d) and 9(e) thevonMises stress owns the samedistribution pattern and tendsto be steady far from the shaft wall

8 Mathematical Problems in Engineering

Table 4 Optimum results of the boundary load unit MPa

1198751

1198752

1198753

1198754

1198755

1198756

1198757

1198758

1198759

11987510

11987511

11987512

2639 1999 75277 2113 2275 9216 5328 12053 1484 1021 5257 2977

2000

4000

6000

8000

10000

12000

14000

4000 6000 8000 100002000

minus24

minus38

minus43

minus43

minus43

minus38

minus43

minus43

minus33

minus38

minus28

minus33

minus38minus43

minus25

minus26

minus27

minus28

minus29

minus30

minus31

minus32

minus33

minus34

minus35

minus36

minus37

minus38

minus39

minus40

minus41

minus42

minus43

minus44

minus45

minus46

minus47

minus48

Figure 8 Distribution of the minimum principal stress and orien-tation

As shown in Figure 10 just like the von Mises stress theshearing stress distribution is in an oval shape near the shaftwall and changes to a butterfly-like shape from the shaft wallThe shearing stress on the shaft wall is maximum (291MPa)due to the stress concentration and it decreases rapidly awayfrom the shaft wall with 89MPa when 06m away Shearingstress tends to be stable when it is 16m away from the shaftwall

From the stress analysis above we can draw the conclu-sion as the following

The stress concentration phenomenon does exist aroundthe hole The von Mises stress and shearing stress aremaximum on the shaft wall The stress distribution is in anoval shape around the shaft wall while it is in a butterfly-likeshape near the hole Both stresses decrease rapidly when theyare 1m away from the shaft wall and tend to be stable when13m away from the shaft wall

7 The Hole Stability Analysis of UGS

71 The Rock Fracture Analysis around the Hole of UGSBased on the Griffith Principle Griffith defined the inherentdefects or micro-fracture in the material as the Griffithfracture Based on theGriffith criterion [22] the rock fracturecondition in the planar stress state is shown below

When (1205901+ 31205902) gt 0 the fracture criterion is

(1205902minus 1205901)2= 8 (120590

2+ 1205901) 120590119905 (20)

Equation (20) can also be like 120590119905= (1205902minus 1205901)28(1205902+ 1205901)

cos 2120573 = (1205901minus 1205902)2(1205902+ 1205901)

When (1205901+ 31205902) le 0 the fracture criterion is

1205902= minus120590119905

sin 2120573 = 0

(21)

where 120590119905is the tensile stress from the rock unidirectional

tensile test 120573 is the angle between the long-axis of theoval fracture and the principal compressive stress axis Thefracture angle is 120579 = 120573

If a new fracture occurs it will be in the direction of ovalfractureThe angle between the new fracture and the originalone is 120595 (Figure 11) The following equation can be obtainedby the fracture criterion

120595 = minus2120573 (22)

Equation (22) shows that the angle between the newfracture and the original one is 2120573 in the condition of (120590

1+

31205902) gt 0 The negative sign indicates the direction of the

new fracture is clockwise rotation by 120573 from the principalcompression stress axis (Figure 11)

For the three-dimensional problem Griffith fracturecriterion can be depicted as follows

When (1205901+ 31205903) gt 0 the fracture criterion is

(1205901minus 1205902)2+ (1205902minus 1205903)2+ (1205903minus 1205901)2

= 24120590119905(1205901+ 1205902+ 1205903)

cos 2120579 =1205901minus 1205903

2 (1205901+ 1205903)

(23)

When (1205901+ 31205903) le 0 the fracture criterion is

120590119905= minus1205903 (24)

Based on the fracture criterion above it can be judgedonly whether the tensile fracture or the shear fracture occurswhile the developing degrees of the fracture cannot bedetermined In fact with the asymmetry of the tectonic stressdistribution the fracture developing degrees of tectonic partdiffer The concept of fracture factor is introduced to judgethe fracture developing degrees of the rock quantitativelyThefracture factor is given by

119868120590=

100381610038161003816100381610038161003816100381610038161003816

120590119905

[120590119905]

100381610038161003816100381610038161003816100381610038161003816

or 119868119905=

100381610038161003816100381610038161003816100381610038161003816

120591119899

[120591119899]

100381610038161003816100381610038161003816100381610038161003816

(25)

Mathematical Problems in Engineering 9

568

568

281281

(a) 000m around the shaft wall

171

171

4343

(b) 060m around the shaft wall104

104

3737

(c) 105m around the shaft wall

85

85

5858

(d) 165m around the shaft wall

78

78

6767

(e) 255m around the shaft wall

Figure 9 von Mises stress distribution of the shaft wall in well UGS1

10 Mathematical Problems in Engineering

291

291

125125

(a) 00m around the shaft wall

89

89

1515

(b) 060m around the shaft wall

57

57

2323

(c) 105m around the shaft wall

47

47

3434

(d) 165m around the shaft wall

44

44

3838

(e) 255m around the shaft wall

Figure 10 The maximum shearing stress distribution around the shaft wall in UGS1

Mathematical Problems in Engineering 11

Table 5 Hole rock tensile fracture analysis results in UGS

Well Maximum principalstressMPa

VerticalstressMPa

Minimum principalstressMPa

PorepressureMPa

TensilestressMPa

Tensile strength[120590119905]MPa

Tensile fracturefactor 119868

120590

UGS6 minus2169 minus2597 minus2793 1200 095 497 0191UGS7 minus2132 minus2610 minus2602 1200 093 484 0192UGS8 minus2121 minus2503 minus2725 1200 099 492 0201UGS9 minus2066 minus2310 minus2783 1200 092 494 0186UGS10 minus1906 minus2515 minus2215 1200 098 495 0198

Table 6 Hole rock shear fracture analysis results in UGS

Well Maximum principalstressMPa

VerticalstressMPa

Minimum principalstressMPa

CohesivestressMPa

Maximum shearingstressMPa

Shear strength[120591119899]MPa

Shear fracturefactor 119868

119905

UGS6 minus2169 minus2597 minus2793 466 249 354 0703UGS7 minus2132 minus2610 minus2602 435 222 368 0603UGS8 minus2121 minus2503 minus2725 405 227 372 0510UGS9 minus2066 minus2310 minus2783 462 213 347 0614UGS10 minus1906 minus2515 minus2215 450 269 357 0754

The newfracture

The newfracture

x

y

120595 = minus2120573

120595 = minus2120573

120573

12059011205901

1205902

1205902

Figure 11 Fracture expansion direction of the oval fracture

where 119868120590and 119868119905are tensile fracture factor and shear fracture

factor 120590119905and 120591

119899are tensile stress and shearing stress in

the fracture plane [120590119905] and [120591

119899] are tensile strength and

shearing resistance The shaft rock will not be cracked whenthe fracture factor is below 1

72 Rock Fracture Analysis Results in the UGS According tothe Griffith fracture criterion presented above we conductthe tensile fracture analysis on the rock of the sidewall inUGS of a Turkey salt cavern With the obtained maximumand minimum horizontal in situ stresses of the test wells

r

120590h1

120590h2

120590h2

120590h1

120590120579120591r120579

120590r

Figure 12 The calculation model of wellbore stress

the tensile stress and shearing stress can be calculated bythe classical elastic theory The calculation model is shownin Figure 12 where 120590

ℎ1 120590ℎ2are the maximum and minimum

horizontal principal in situ stresses 120590119903is the radial normal

stress 120590120579is the tangential normal stress and 120591

119903120579is the

shearing stressThe results are shown in Table 5 showing that the

maximum tensile stress is smaller than the tensile strengthTherefore on the rock of the shaft wall no fracture will bedeveloped

Moreover according to the Coulomb-Navier failure cri-terion the shear fracture analysis is conducted on the rock ofthe shaft wall in UGS of a Turkey salt cavern Table 6 showsthe analysis results The shear fracture factors 119868

119905 in the hole

are all under 1 Given that themaximum shearing stress is lessthan the shearing strength the rock of the shaft wall will notdevelop a shear fracture

12 Mathematical Problems in Engineering

8 Conclusion

In this paper we improved the nonuniqueness of the in situstress field inversion solution by a priori constraints Fromthe perspective of basic equations of elastic mechanics anin situ stress numerical inversion model is built and basicfactors of in situ stress field are considered in applying thefinite element inversion method on the boundary loads ofthe in situ stress field An appropriate geometrical modeland suitable boundary conditions ensured an accurate in situstress field is derived The method is successfully applied inUGS of a Turkey salt cavern The optimisation results onprincipal stress are close to those from measurement By theoptimisation the overall in situ stress field of the researcharea is obtained and the stress distribution of the shaft wallis presented according to the calculation results The safetyof the shaft wall is confirmed by applying the correspondingfracture criterion

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to acknowledge the financial support ofthe National Natural Science Foundation of China (Grantsnos 51374228 51274231 and U1262208) and the CNPC KeyLaboratory Fund (Contract no 2014A-4214)

References

[1] B S Aadnoy ldquoInversion technique to determine the in-situstress field from fracturing datardquo Journal of Petroleum Scienceand Engineering vol 4 no 2 pp 127ndash141 1990

[2] M Bloch C A M Siqueira F H Ferreira and J C JConceicao ldquoTechniques for determining in-situ stress directionand magnitudesrdquo in Proceedings of the Latin American andCaribbean Petroleum Engineering Conference SPE 39075-MSRio de Janeiro Brazil August-September 1997

[3] J Huang D V Griffiths and S-W Wong ldquoIn situ stress deter-mination from inversion of hydraulic fracturing datardquo Interna-tional Journal of Rock Mechanics and Mining Sciences vol 48no 3 pp 476ndash481 2011

[4] J Djurhuus and B S Aadnoslashy ldquoIn situ stress state from inversionof fracturing data from oil wells and borehole image logsrdquoJournal of Petroleum Science and Engineering vol 38 no 3-4pp 121ndash130 2003

[5] B S Aadnoy ldquoIn-situ stress directions from borehole fracturetracesrdquo Journal of Petroleum Science and Engineering vol 4 no2 pp 143ndash153 1990

[6] C A Barton D Moos P Peska and M D Zoback ldquoUtilizingwellbore image data to determine the complete stress tensorapplication to permeability anisotropy and wellbore stabilityrdquoThe Log Analyst vol 38 no 6 pp 21ndash33 1997

[7] H Zhao F Ma J Xu and J Guo ldquoIn situ stress field inversionand its application in mining-induced rock mass movementrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 53 pp 120ndash128 2012

[8] Z D Chen Q A Meng and T F Wan ldquoNumerical simulationof tectonic stress field in Gulong depression in Songliao Basinusing elasitic-plastic increment methodrdquo Earth Science Fron-tiers vol 9 no 2 pp 483ndash491 2002 (Chinese)

[9] D K Singha and R Chatterjee ldquoGeomechanical modelingusing finite element method for prediction of in-situ stress inKrishna-Godavari basin Indiardquo International Journal of RockMechanics and Mining Sciences vol 73 no 5 pp 15ndash27 2015

[10] G Maier R Ardito and R Fedele ldquoInverse analysis problemsin structural engineering of concrete damsrdquo ComputationalMechanics vol 9 pp 5ndash10 2004

[11] G Maier M Bocciarelli G Bolzon and R Fedele ldquoInverseanalyses in fracture mechanicsrdquo International Journal of Frac-ture vol 138 no 1ndash4 pp 47ndash73 2006

[12] F Yin and D L Gao ldquoMechanical analysis and design of casingin directional well under in-situ stressesrdquo Journal of Natural GasScience and Engineering vol 20 pp 285ndash291 2014

[13] S Sakurai and S Akutagawa ldquoReport some aspects of backanalysis in geotechnical engineeringrdquo in Proceedings of theISRM International Symposium (EUROCK rsquo93) pp 1133ndash1140Lisboa Portugal June 1993

[14] L Z Xue Engineering Optimum Programming TechnologiesTianjing University Press Tianjing China 1998 (Chinese)

[15] F Li J-A Wang and J C Brigham ldquoInverse calculation of insitu stress in rock mass using the Surrogate-Model AcceleratedRandom Search algorithmrdquo Computers and Geotechnics vol 61pp 24ndash32 2014

[16] X Z Wang W Q Liu Y G Sun and Y J Ma ldquoField testingof fracturing techanolgy in coal formed gas wellrdquoOil Drilling ampProduction Technology vol 23 no 2 pp 58ndash61 2001 (Chinese)

[17] J Teng Y B Yao D M Liu and Y D Cai ldquoEvaluation ofcoal texture distributions in the southern Qinshui basin NorthChina investigation by amultiple geophysical loggingmethodrdquoInternational Journal of Coal Geology vol 140 pp 9ndash22 2015

[18] Z Meng J Zhang and R Wang ldquoIn-situ stress pore pressureand stress-dependent permeability in the Southern QinshuiBasinrdquo International Journal of Rock Mechanics and MiningSciences vol 48 no 1 pp 122ndash131 2011

[19] P Lancaster and K Salkauskas ldquoSurfaces generated by movingleast squaresmethodsrdquoMathematics of Computation vol 37 no155 pp 141ndash158 1981

[20] AHMedley ldquoStorage of natural gas in salt cavernsrdquo in Proceed-ings of the SPE Northern Plains Section Regional Meeting SPE-5427-MS Conference Paper Society of Petroleum EngineersOmaha Neb USA May 1975

[21] G Han M S Bruno K Lao J Young and L Dorfmann ldquoGasstorage and operations in single-bedded salt caverns stabilityanalysesrdquo SPEProduction andOperations vol 22 no 3 pp 368ndash376 2007

[22] P Berest and B Brouard ldquoLong-term behavior of salt cavernsrdquoin Proceedings of the 48th US Rock MechanicsGeomechanicsSymposium vol 14 Minneapolis Minn USA June 2014ARMA-2014-7032

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

multiple fracturing and induced fractures from imaginglogs to calculate the in situ stress state The magnitudeinformation is mainly calculated from fracturing data whilethe orientation information can be obtained from image logs

However measurement is difficult to carry out becauseof high cost on an injection-production well Thereforeseveral in situ stress values can only be obtained throughexperiments The stress distribution of the test zone can becalculated quicklywith the help of finite element optimisationand the results can be more accurate with some optimi-sation methods implemented To carry out the inversioncalculation we need to construct the finite element modelwith proper boundary conditions and loads Some minormodifications should also be incorporated into the modelto eliminate the effect of stress concentration which will bediscussed below

Boundary conditions and loading methods cannot beinput directly because the tectonic movement is unknownand the geologic structure is complicated These circum-stances make in situ stress field analysis difficult to carryout So boundary load inversion comes to be the crucialway to conduct research on the in situ stress field Fromthe viewpoint of geomechanics the tectonic stress field isthe in situ stress field forming the tectonic system and typeincluding its distribution area and inside stress distributionDuring the analysis rock density could be determined rel-atively accurately in a small variation range with differentprecision levels of gravity stress and tectonic stress Instead ofbeing inversed based on terrain conditions the finite elementmethod could be used to calculate gravity stress to ensure theprecision can satisfy the engineering requirement Howevertectonic stress field calculation is different [7] During theconstruction of UGS the distribution of the in situ stress fieldalong the plane at the top of the salt cavern is needfulwhile thein situ stress information of only a fraction ofwells in a certainarea is availableThen inversion and extrapolation need to beperformed to obtain the stress field of the whole area Amongthe present numerical simulations and analysis methods ofthe in situ stress field the most widely used approach iscut-and-try and adjustment of boundary loads [8ndash13] Giventhe complexity of the geological structure and the limitedreference of the in situ stress boundary conditions andloading methods become the difficult problem in calculationand analysis part of a numerical simulation The major issueis how to improve the precision of the inversion and reliabilityof the data Simulation is an effective way to take terrain andgeological features into consideration And only by this canrelevant theoretical or numerical analyses such as back anal-ysis back calculation and simulation be performed to predictthe tectonic stress and its mode based on the measureddata [12]The boundary loads identification and optimisationmethod presented by this paper adopts damped least squaresalgorithm to approximate the loads by setting parametersfrom stress testing references The initial value for iterationis got by the unit load method to avoid the limits from thestructure geometric features and boundary conditions Bycontrolling the error of observing point the identification onuncertain and unknown boundary loads of the in situ stressfield is achieved A few ways are attempted and compared to

improve the data reliability and the precision of in situ stressfield inversion calculation in this paper

2 Building the Finite Element Model

It is difficult to achieve an analytical solution of boundaryloads for a formation due to the complexity of the geolog-ical structure and boundary conditions The finite elementmethod is employed here for mechanical analysis to makesure that the research method can have a general application

Though salt is viscoelastic aUGS reservoir stress field canbe treated as an elastic problem to simplify the analysis Thisstudy focuses on the original rock salt without consideringthe effect of creep and relax The linear elastic assumptionmay bring some deviation with the actual situation but it ishigh needed for figuring out a specific engineering problem

The applied loads generate displacement field stress fieldand strain field inside for a certain area The basic equationsof small deformation elastic mechanics are as follows

equilibrium differential equation

119871119879120590 + 119876 = 0 (1)

geometric equation

120576 = 119871119880 (2)

physical equation

120590 = 119863120576 (3)

where 119871 is the differential operator and119863 is the elastic matrixdepending on the elastic modulus 119864 of the formation andPoissonrsquos ratio ]

Generally the finite element formula is

119870119880 = 119865 (119883) (4)

where119870 is the global stiffness matrixU is the node displace-ment matrix 119865(119883) is the boundary load array and 119883 is theparameter vector used to describe the boundary loads of UGSreservoir

Equation (4) is transformed to

119880 = 119870minus1119865 (119883) (5)

Equation (2) is substituted in (3) to get the followingequation

120590 = 119863119871119880 (6)

Then (5) is substituted in (6) that is

120590 = 119863119871119870minus1119865 (119883) (7)

From (7) the nodal stress and the boundary loads satisfythe following equation

120590 = 119879119865 (119883) (8)

Mathematical Problems in Engineering 3

where119879 = 119863119871119870minus1119879 is the transfermatrix between the nodal

stress matrix and the boundary loads array For the linearelastic body 119879 could be determined when the finite elementmodel of UGS reservoir is established 119865(119883) is uncertainbecause the loads are unknown and only a part of120590 is certainsince the stress values of only a few wells in the reservoir tobe studied are known Therefore in general the boundaryloads parameter vector 119883 cannot be calculated from (8) andis waiting to be discovered

3 Optimisation Method

In situ stress field inversion is used to find the model orparameters that can fit actual data under certain principlesIt is not only a method of data fitting but also a way ofoptimising The inversion result is closely related to theoptimal criterions which is used to evaluate the degree ofdata fitting Sometimes we have prior information aboutsome areas that have already been researched Based on thisprior information we could restrain the optimal criterion andparameters Generally the data going to be used in stressinversion calculation is obtained from the reservoir stressfield and contains the stress of logging fracturing boreholebreakout or core testing of certain wells The principle ofstress inversion (or objective function) could be described asthe followingmodels by using the least squaresmethod as theoptimal criterion

(1) Constraint Model If the stress value of the test well is 120590119871

that is the simulation value of the stress field 120590 from the finiteelement calculation is known the model could be describedas

min Φ (119883) = (120590 minus 120590119871)119879(120590 minus 120590

119871)

st ℎ1le 120593 (119883) le ℎ

2

119886119894le 119883119894le 119887119894

(9)

where 120590 is derived from the finite element calculation Φ(119883)

is the mean squared error between the measurement andcalculation 119883 is the unknown boundary load vector andℎ1and ℎ

2are the boundary load parameter functions of the

upper bound and lower bound respectively determined byprior information 119886

119894and 119887119894are the boundary load parameters

of upper and lower bound respectively

(2) Unconstraint Model An unconstraint model exists whenthe condition in (9) is not met that is

minΦ (119883) = (120590 minus 120590119871)119879(120590 minus 120590

119871) (10)

where symbols are the same as those in (9)

4 Inversion Algorithm

According to the principle of inversion calculation backanalysis is used to calculate the input data of a system byanalysing its output data [13] Therefore if the system is wellconstructed it is possible to obtain the input data from its

output data which is usually achieved through experimentsThe back analysis ensures that a unique solution can begenerated as long as the mechanical model is real and thedata is sufficient

The general way of inversion is to present the parametervector of boundary load 119883 and then calculate the reservoirrsquosstress response value under 119883 During this process thefinite element method is used to approach the actual data ofsome well sites based on the least squares method Usuallythe Gauss-Newton method is adopted to solve the leastsquares problem of (9) or (10) However ill-conditionedcoefficient matrix of the equation low iterative speed andspurious convergence may occur sometimes In this paper adamped least squares method [Levenberg-Marquardt (LM)method] [14] is implemented to approach the parametervector of boundary load 119883 The LM method which has ahigh optimisation speed and precision possesses both theadvantages of the gradient method and the Newton methodWhen the initial value of damping factor 120582

119896is small the step

size is set to be equal to that of the Newton method while itis equal to that of the gradient descent when 120582

119896is large

When the iteration reaches step k the column vector ofstress response and boundary load120590

119896 and119883119896 are setThen120590119896

series expansion around119883119896 is performed and the trace above

the second-order polynomial is ignored to get the followingequation [15]

120590119896= 120590119896minus1

+120597120590

120597119883(119883119896minus 119883119896minus1

)

= 120590119896minus1

+ 119869 sdot (119883119896minus 119883119896minus1

)

(11)

where 119869 is the Jacobi matrix of 120590119896 in119883119896 and it is defined from

the finite difference method as

119869119894119895=

120597120590119894

120597119883119895

asympΔ120590119894

Δ119883119895

=

120590119894(1198831 1198832 119883

119895+ Δ119883119895 119883

119899)

Δ119883119895

minus

120590119894(1198831 1198832 119883

119895 119883

119899)

Δ119883119895

(12)

where 120590119894could be calculated from (7)

The search direction used in LM method is from thesolution of a set of linear equalities The iterative formula is

[119869119879(119883(119896)

) 119869 (119883(119896)

) + 120582119896119868] Δ119883

(119896)

= minus119869119879(119883(119896)

) sdot 120590119896minus1

minus 120590119871

(13)

where Δ119883 is the search direction vector of 119883 and 119868 is theunit matrix Scalar 120582

119896is the damping factor controlling the

direction and value ofΔ119909119896When 120582

119896is 0 the direction ofΔ119909

119896

is the same as the result of the Gauss-Newton method When

4 Mathematical Problems in Engineering

120582119896tends to infinity we could derive the following equation

from (13)

[119869119879(119883(119896)

) 119869 (119883(119896)

) + 120582119896119868] Δ119883

(119896) asymp 120582119896119868 Δ119883

(119896)

= minus119869119879(119883(119896)

) sdot 120590119896minus1

minus 120590119871

(14)

Thus

Δ119883(119896)

= minus1

120582119896

119869119879(119883(119896)

) sdot 120590119896minus1

minus 120590119871 (15)

where Δ119883(119896) is in the direction of the zero vector and has a

steep descending slope for 120582119896 which is large enough that is

Φ(119909(119896)

+ Δ119909(119896)

) lt Φ (119909(119896)

) (16)

From (13) we derived

Δ119883(119896)

= minus [119869119879(119883(119896)

) 119869 (119883(119896)

) + 120582119896119868]minus1

119869119879(119883(119896)

)

sdot 120590119896minus1

minus 120590119871

(17)

Then we got

119883(119896)

= 119883(119896minus1)

+ Δ119883(119896)

(18)

The boundary load vector could be derived from iterativesolving of (13)ndash(18) The key step is how to choose iterativeinitial value If the range of the boundary load is knownwe can set the iterative initial value of 119883 as the average ofthe upper and lower bounds If the range is an unboundeddomain the iterative initial value of 119883 could be determinedafter experiential trials

From the above discussion if the measured data ofreservoir stress is available the boundary load could be deter-mined The load parameter vector and reservoirrsquos horizontalstress 120590

119871could satisfy

119866119883 = 120590119871 (19)

where 119866 isin 119862119898times119899 denotes the transfer matrix 119898 is the point

number for the measurement and 119899 is the dimension of theboundary load vector If 119899 gt 119898 a set of boundary loadvectors could be clear based on the principle of controlling thecomprehensive error to be minimum for the nonuniquenessof the solution

The specific steps of finite element inversion of the in situstress field go as follows Firstly choose a proper objectivefunction Secondly adjust and search parameters by theoptimal method and forward-model the computed value bythe finite element numerical algorithm Thirdly substitutethe measured and calculated values into the objective func-tion and judge the result from the objective function Ifthe result did not reach the minimum the adjusting andsearching should be continued Repeat this process untilthe result from objective function reaches the minimumvalue and finally the boundary load parameter is generatedAfterwards the forward modelling is carried out to acquirethe reservoirrsquos stress field by the boundary load derived

Initial data X

Parameter modification

Inversion parameter and results

Feature analysis of in situ stress field

Forward numerical calculation 120590 = TF(X)

Objective function 120601(X)

Measured data 120590L

120601(X) le 120576

Figure 1 Flowchart of the finite element inversion

Object

P1

P2

P3

P4

P5

P6

P7

P8

P9

P10

P11

P12

Figure 2 Computational model for the boundary load

from the inversion (Figure 1 shows the inversion process)The applied boundary condition is shown in Figure 2 Thecalculation is performed in the entire circle area to avoidstress concentration Assuming the strata are boundless andstress is continuously changed if the stress applied on thecircle can make the inversion results close enough to the testresults the stress applied on the rectangular must be rightThe stress distribution in the object area is the result we wantfrom the inversion

5 Example and Application

Based on the aforementioned theory in order to get theboundary load of the in situ stress field we could programthe finite element iteration by finite element software Inthis paper a Turkey salt cavern is chosen to illustrate theapplication of reversion technology on resolving in situ stressboundary load Furtherly this method is also applied on thein situ stress analysis in Qinshui basin of China [16ndash18] Theleast moving squares method can also be used to deduce

Mathematical Problems in Engineering 5

1150 1100 1150 1150 1200 1200 1250 125012501100

1250

Tuz formation

Cihanbeyli to Kochisar formation

1100

1800

1600

1400

1200

1000

800

600

400

200

0

Dep

th b

elow

surfa

ce (m

)

UG

S12

UG

S11

UG

S10

UG

S1

UG

S2

UG

S9

UG

S8

UG

S7

UG

S6

UG

S5

UG

S4

UG

S3

Figure 3 Section map of UGS in Turkey salt cavern

UGS11

UGS10

UGS1

UGS2UGS9

UGS8

UGS7

UGS6

UGS5 UGS4UGS3

Figure 4 Three-dimensional overview of UGS in Turkey salt cavern

the in situ stress field by carefully selecting the primaryfunction and the weight function [19] The UGS distributionis shown in Figures 3ndash5 Figure 3 shows the well depths ofmultiple UGS wells The minimum depth is 1110m whilethe maximum depth is 1500m Figure 4 shows the three-dimensional distribution of these UGS wells Figure 5 showsthe planar structure of the salt cavern Twelve reservoirs asmany as UGS wells construct the salt cavern All sites are inthe southeast section which is indicated by the blue spotsThe red line in Figure 5 indicates the fault zone of the researcharea

51 Building UGS in a Salt Cavern Salt owns several proper-ties that make it ideal for gas storage [19] It has a moderatelyhigh strength and flows plastically to close fracture that couldotherwise become a leak The porosity and permeability ofsalt to liquid and gaseous hydrocarbons are close to zerothereby preventing stored gas escape [20 21] Salt caverns canprovide high deliverability and gas can be withdrawn quicklybecause no pressure from flow is lost through the pores Asalt cavern in south Turkey is used to construct undergroundnatural gas storage to compensate season-related fluctuations

of gas consumption Moreover by a salt cavern gas can beprovided to customers under optimum economic conditionsbetween buying and selling To construct UGS by a saltcavern a well should be drilled first and then water shouldbe injected to dissolve the salt Brine is pumped out of the saltcavern and a cavity is formed which can be treated to storenatural gas

52 Calculation Model and Boundary Conditions Based onthe prerequisite of the numerical simulation a geologicmodel the target layer isolation body of a rock block is takenas the simulation object Two rules should be followed todetermine the computational area (1) extend the geometricrange to eliminate or weaken the effects from the bound-ary conditions on the research area and (2) the geometryconstraint conditions on the boundary are easy to get Ingeneral the layered distribution of underground sedimentaryrock could be treated as a plane problem of elastic mechanicsin the comprehensive mathematical model Figure 5 is themechanical model of the research area which is a rectangulararea with a length of 8 km and width of 32 km Three majorfault zones are taken into consideration The lengths and

6 Mathematical Problems in Engineering

UGS11UGS12

UGS1UGS10

UGS2 UGS9UGS8 UGS7

UGS6 UGS5UGS4 UGS3

0 1000 2000(m)

Figure 5 Planar structure of the research area and the distributionof the UGS

widths of fault zones are 8 km lowast 1 km 12 km lowast 08 km and35 km lowast 08 km respectively

The following eight factors are taken as the basic factorsof tectonic force compressive or tensile forces and shearforces in the east west south and north The boundaryload is adjusted depending on the size of research area Asthe stress field is continuously changed the external stressin the circle can be divided into 12 parts which is largeenough to guarantee the accuracy of calculation By changingthe value of the 12 boundary forces we could simulate thecompressive or tensile forces in the direction of eastndashwest andsouthndashnorth as well as shear deformation in the horizontalplane to get the in situ stress field distribution of the partialresearch area At the same time boundary force constraintis applied to make the value of the resultant force and theresultant moment zero (Figure 6 shows the method to applythe boundary force)The edge points of the lower boundary ofthemodel are fixed to ensure no rigid translation and rotationin the research area

Generally a geologic body is taken as a rock with auniform block to confirm the rock physical parameters fromdifferent tectonic positions The basic assumption is thateach fault zone is homogeneous The rock strength Youngrsquosmodulus and other rock mechanical parameters should bedecreased according to the rock properties in specific depth(Table 1 shows information about mechanical parameterschosen by the model) The cored sample is taken from the

1ElementsMAT Num

Figure 6 Mechanical model of the calculated area

Table 1 Rock mechanics parameters

Number Media type Elasticity modulusMPa Poissonrsquos ratio ]1 Reservoir 31200 02252 Fault zone 1 22500 02703 Fault zone 2 23500 02604 Fault zone 3 24000 02905 Other layers 35200 0220

spot field and by indoor experiment elasticity modulus andPoissonrsquos ratio are obtained The grids of the model adoptedmainly are four-node isoperimetric rectangular elements andfor part of the area triangle elements are used In additionthe element length of the fault zone and its surrounding areais shorter and the grids are relatively finer than those of otherparts

53 Inversion of the In Situ Stress Field of the Salt Cavern

(1) Initial boundary conditions the initial values of thedistributed normal stress 119875

1sim 11987512in the east south

west and north could be set as an arbitrary value with10MPa here (Figure 6 shows how the boundary forceacts)

(2) Constraint object based on the reference about insitu stress measurement we chose 10 values of in situstress and five directions from eight survey points asknown constraint parameters to inverse 12 unknownboundary forces (Figure 5 shows the distribution ofconstraint points and measured data are shown inTables 2 and 3)

(3) Searching range the range of the boundary nor-mal stress constraint value is 1MPasim30MPa (lowerboundsimupper bound) and in the same way therange of the boundary shear stress constraint valueminus10MPasim10MPa The precision of the iteration ter-mination is 10minus5MPa2The boundary force constraintshould be taken into consideration to make sure thatthe resultant moment of the boundary force is zero

Mathematical Problems in Engineering 7

Table 2 Comparison between the inversion results and the measured values on principal stress magnitude

Measured points Stress components Inversion valueMPa Measured valueMPa Relative error

UGS6 120590119867

minus2169 minus2323 66120590ℎ

minus2793 minus2777 058

UGS7 120590119867

minus2132 minus1996 681120590ℎ

minus2602 minus261 031

UGS8 120590119867

minus2121 minus2026 469120590ℎ

minus2725 minus2852 445

UGS9 120590119867

minus2066 minus2226 719120590ℎ

minus2783 minus2664 447

UGS10 120590119867

minus1906 minus1856 269120590ℎ

minus2515 minus2385 545

Table 3 Comparison between the inverted results and themeasuredvalues on the direction of maximum principal stress

Measuredpoints Inversion value(∘) Measured value(∘) Error(∘)

UGS6 NE57327 NE55 2327UGS7 NE51798 NE535 minus1702

UGS8 NE52334 NE549 minus257

UGS9 NE4132 NE451 minus378

UGS10 NE51713 NE536 minus1887

The relevant constrained inversion model could bederived from (9)

(4) Inversion result based on the conditions above thecalculation results on the survey points t from inver-sion are shown in Tables 2 and 3 where 120590

119867and

120590ℎare the maximum and minimum principal stress

magnitude respectively The measurement resultsfrom well logging are also presented and comparedwith the inversion results Table 4 shows the results ofthe boundary load inversion

Tables 2 and 3 show that the maximum relative errorsof stress components 120590

119867and 120590

ℎare controlled within 8

The errors of the direction of maximum principal stress arecontrolled within 4∘ All the stress values obtained from theinversion reached or are close to the actual data and theprecision of the inversion results could meet engineeringrequirements

We applied the optimised boundary force from the inver-sion to perform forward calculation This approach couldlead to the distribution of the stress field value and directionof the Turkey salt cavern (shown in Figures 7 and 8)

6 Stress Distribution Character of the Wellnear the Hole of the UGS

On the basis of the analysis above the two principal stresses120590119867and 120590ℎof the 12 wells are determined Given that the stress

concentration occurs by the hole which will influence thestability of the shaft wall the von Mises stress distributioncharacter near the shaft wall (0ndash255m) should be checked

minus29

minus13minus14

minus15

minus16

minus17

minus18

minus19

minus20

minus21

minus22

minus23

minus24

minus25

minus26

minus27

minus28

minus29

minus30

minus31

minus32

minus33

minus34

minus24

minus19

minus19

minus19

minus19

minus19minus19

2000

4000

6000

8000

10000

12000

14000

4000 6000 8000 100002000

Figure 7 Distribution of the maximum principal stress and orien-tation

up as shown in Figure 9 Here well UGS1 is taken as anexample with well diameter of UGS1 279 cm bottom holepressure 12MPa overburden pressure 231MPa and welldepth 1100m

As shown in Figure 9 given that stress nonhomogeneouswith the in situ stress exists the von Mises stress distributionof the shaft wall is in an oval shape The von Mises stresshas a higher value in the shaft wall which is within 281ndash568MPa The von Mises stress decreases rapidly away fromthe shaft wall and the shape changes to a butterfly-like oneThe maximum von Mises stress decreases to 171MPa when06m away from the shaftwall It is worthmentioning that theresult here contains the influence from stress concentrationaround the shaft wall As shown in Figures 9(d) and 9(e) thevonMises stress owns the samedistribution pattern and tendsto be steady far from the shaft wall

8 Mathematical Problems in Engineering

Table 4 Optimum results of the boundary load unit MPa

1198751

1198752

1198753

1198754

1198755

1198756

1198757

1198758

1198759

11987510

11987511

11987512

2639 1999 75277 2113 2275 9216 5328 12053 1484 1021 5257 2977

2000

4000

6000

8000

10000

12000

14000

4000 6000 8000 100002000

minus24

minus38

minus43

minus43

minus43

minus38

minus43

minus43

minus33

minus38

minus28

minus33

minus38minus43

minus25

minus26

minus27

minus28

minus29

minus30

minus31

minus32

minus33

minus34

minus35

minus36

minus37

minus38

minus39

minus40

minus41

minus42

minus43

minus44

minus45

minus46

minus47

minus48

Figure 8 Distribution of the minimum principal stress and orien-tation

As shown in Figure 10 just like the von Mises stress theshearing stress distribution is in an oval shape near the shaftwall and changes to a butterfly-like shape from the shaft wallThe shearing stress on the shaft wall is maximum (291MPa)due to the stress concentration and it decreases rapidly awayfrom the shaft wall with 89MPa when 06m away Shearingstress tends to be stable when it is 16m away from the shaftwall

From the stress analysis above we can draw the conclu-sion as the following

The stress concentration phenomenon does exist aroundthe hole The von Mises stress and shearing stress aremaximum on the shaft wall The stress distribution is in anoval shape around the shaft wall while it is in a butterfly-likeshape near the hole Both stresses decrease rapidly when theyare 1m away from the shaft wall and tend to be stable when13m away from the shaft wall

7 The Hole Stability Analysis of UGS

71 The Rock Fracture Analysis around the Hole of UGSBased on the Griffith Principle Griffith defined the inherentdefects or micro-fracture in the material as the Griffithfracture Based on theGriffith criterion [22] the rock fracturecondition in the planar stress state is shown below

When (1205901+ 31205902) gt 0 the fracture criterion is

(1205902minus 1205901)2= 8 (120590

2+ 1205901) 120590119905 (20)

Equation (20) can also be like 120590119905= (1205902minus 1205901)28(1205902+ 1205901)

cos 2120573 = (1205901minus 1205902)2(1205902+ 1205901)

When (1205901+ 31205902) le 0 the fracture criterion is

1205902= minus120590119905

sin 2120573 = 0

(21)

where 120590119905is the tensile stress from the rock unidirectional

tensile test 120573 is the angle between the long-axis of theoval fracture and the principal compressive stress axis Thefracture angle is 120579 = 120573

If a new fracture occurs it will be in the direction of ovalfractureThe angle between the new fracture and the originalone is 120595 (Figure 11) The following equation can be obtainedby the fracture criterion

120595 = minus2120573 (22)

Equation (22) shows that the angle between the newfracture and the original one is 2120573 in the condition of (120590

1+

31205902) gt 0 The negative sign indicates the direction of the

new fracture is clockwise rotation by 120573 from the principalcompression stress axis (Figure 11)

For the three-dimensional problem Griffith fracturecriterion can be depicted as follows

When (1205901+ 31205903) gt 0 the fracture criterion is

(1205901minus 1205902)2+ (1205902minus 1205903)2+ (1205903minus 1205901)2

= 24120590119905(1205901+ 1205902+ 1205903)

cos 2120579 =1205901minus 1205903

2 (1205901+ 1205903)

(23)

When (1205901+ 31205903) le 0 the fracture criterion is

120590119905= minus1205903 (24)

Based on the fracture criterion above it can be judgedonly whether the tensile fracture or the shear fracture occurswhile the developing degrees of the fracture cannot bedetermined In fact with the asymmetry of the tectonic stressdistribution the fracture developing degrees of tectonic partdiffer The concept of fracture factor is introduced to judgethe fracture developing degrees of the rock quantitativelyThefracture factor is given by

119868120590=

100381610038161003816100381610038161003816100381610038161003816

120590119905

[120590119905]

100381610038161003816100381610038161003816100381610038161003816

or 119868119905=

100381610038161003816100381610038161003816100381610038161003816

120591119899

[120591119899]

100381610038161003816100381610038161003816100381610038161003816

(25)

Mathematical Problems in Engineering 9

568

568

281281

(a) 000m around the shaft wall

171

171

4343

(b) 060m around the shaft wall104

104

3737

(c) 105m around the shaft wall

85

85

5858

(d) 165m around the shaft wall

78

78

6767

(e) 255m around the shaft wall

Figure 9 von Mises stress distribution of the shaft wall in well UGS1

10 Mathematical Problems in Engineering

291

291

125125

(a) 00m around the shaft wall

89

89

1515

(b) 060m around the shaft wall

57

57

2323

(c) 105m around the shaft wall

47

47

3434

(d) 165m around the shaft wall

44

44

3838

(e) 255m around the shaft wall

Figure 10 The maximum shearing stress distribution around the shaft wall in UGS1

Mathematical Problems in Engineering 11

Table 5 Hole rock tensile fracture analysis results in UGS

Well Maximum principalstressMPa

VerticalstressMPa

Minimum principalstressMPa

PorepressureMPa

TensilestressMPa

Tensile strength[120590119905]MPa

Tensile fracturefactor 119868

120590

UGS6 minus2169 minus2597 minus2793 1200 095 497 0191UGS7 minus2132 minus2610 minus2602 1200 093 484 0192UGS8 minus2121 minus2503 minus2725 1200 099 492 0201UGS9 minus2066 minus2310 minus2783 1200 092 494 0186UGS10 minus1906 minus2515 minus2215 1200 098 495 0198

Table 6 Hole rock shear fracture analysis results in UGS

Well Maximum principalstressMPa

VerticalstressMPa

Minimum principalstressMPa

CohesivestressMPa

Maximum shearingstressMPa

Shear strength[120591119899]MPa

Shear fracturefactor 119868

119905

UGS6 minus2169 minus2597 minus2793 466 249 354 0703UGS7 minus2132 minus2610 minus2602 435 222 368 0603UGS8 minus2121 minus2503 minus2725 405 227 372 0510UGS9 minus2066 minus2310 minus2783 462 213 347 0614UGS10 minus1906 minus2515 minus2215 450 269 357 0754

The newfracture

The newfracture

x

y

120595 = minus2120573

120595 = minus2120573

120573

12059011205901

1205902

1205902

Figure 11 Fracture expansion direction of the oval fracture

where 119868120590and 119868119905are tensile fracture factor and shear fracture

factor 120590119905and 120591

119899are tensile stress and shearing stress in

the fracture plane [120590119905] and [120591

119899] are tensile strength and

shearing resistance The shaft rock will not be cracked whenthe fracture factor is below 1

72 Rock Fracture Analysis Results in the UGS According tothe Griffith fracture criterion presented above we conductthe tensile fracture analysis on the rock of the sidewall inUGS of a Turkey salt cavern With the obtained maximumand minimum horizontal in situ stresses of the test wells

r

120590h1

120590h2

120590h2

120590h1

120590120579120591r120579

120590r

Figure 12 The calculation model of wellbore stress

the tensile stress and shearing stress can be calculated bythe classical elastic theory The calculation model is shownin Figure 12 where 120590

ℎ1 120590ℎ2are the maximum and minimum

horizontal principal in situ stresses 120590119903is the radial normal

stress 120590120579is the tangential normal stress and 120591

119903120579is the

shearing stressThe results are shown in Table 5 showing that the

maximum tensile stress is smaller than the tensile strengthTherefore on the rock of the shaft wall no fracture will bedeveloped

Moreover according to the Coulomb-Navier failure cri-terion the shear fracture analysis is conducted on the rock ofthe shaft wall in UGS of a Turkey salt cavern Table 6 showsthe analysis results The shear fracture factors 119868

119905 in the hole

are all under 1 Given that themaximum shearing stress is lessthan the shearing strength the rock of the shaft wall will notdevelop a shear fracture

12 Mathematical Problems in Engineering

8 Conclusion

In this paper we improved the nonuniqueness of the in situstress field inversion solution by a priori constraints Fromthe perspective of basic equations of elastic mechanics anin situ stress numerical inversion model is built and basicfactors of in situ stress field are considered in applying thefinite element inversion method on the boundary loads ofthe in situ stress field An appropriate geometrical modeland suitable boundary conditions ensured an accurate in situstress field is derived The method is successfully applied inUGS of a Turkey salt cavern The optimisation results onprincipal stress are close to those from measurement By theoptimisation the overall in situ stress field of the researcharea is obtained and the stress distribution of the shaft wallis presented according to the calculation results The safetyof the shaft wall is confirmed by applying the correspondingfracture criterion

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to acknowledge the financial support ofthe National Natural Science Foundation of China (Grantsnos 51374228 51274231 and U1262208) and the CNPC KeyLaboratory Fund (Contract no 2014A-4214)

References

[1] B S Aadnoy ldquoInversion technique to determine the in-situstress field from fracturing datardquo Journal of Petroleum Scienceand Engineering vol 4 no 2 pp 127ndash141 1990

[2] M Bloch C A M Siqueira F H Ferreira and J C JConceicao ldquoTechniques for determining in-situ stress directionand magnitudesrdquo in Proceedings of the Latin American andCaribbean Petroleum Engineering Conference SPE 39075-MSRio de Janeiro Brazil August-September 1997

[3] J Huang D V Griffiths and S-W Wong ldquoIn situ stress deter-mination from inversion of hydraulic fracturing datardquo Interna-tional Journal of Rock Mechanics and Mining Sciences vol 48no 3 pp 476ndash481 2011

[4] J Djurhuus and B S Aadnoslashy ldquoIn situ stress state from inversionof fracturing data from oil wells and borehole image logsrdquoJournal of Petroleum Science and Engineering vol 38 no 3-4pp 121ndash130 2003

[5] B S Aadnoy ldquoIn-situ stress directions from borehole fracturetracesrdquo Journal of Petroleum Science and Engineering vol 4 no2 pp 143ndash153 1990

[6] C A Barton D Moos P Peska and M D Zoback ldquoUtilizingwellbore image data to determine the complete stress tensorapplication to permeability anisotropy and wellbore stabilityrdquoThe Log Analyst vol 38 no 6 pp 21ndash33 1997

[7] H Zhao F Ma J Xu and J Guo ldquoIn situ stress field inversionand its application in mining-induced rock mass movementrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 53 pp 120ndash128 2012

[8] Z D Chen Q A Meng and T F Wan ldquoNumerical simulationof tectonic stress field in Gulong depression in Songliao Basinusing elasitic-plastic increment methodrdquo Earth Science Fron-tiers vol 9 no 2 pp 483ndash491 2002 (Chinese)

[9] D K Singha and R Chatterjee ldquoGeomechanical modelingusing finite element method for prediction of in-situ stress inKrishna-Godavari basin Indiardquo International Journal of RockMechanics and Mining Sciences vol 73 no 5 pp 15ndash27 2015

[10] G Maier R Ardito and R Fedele ldquoInverse analysis problemsin structural engineering of concrete damsrdquo ComputationalMechanics vol 9 pp 5ndash10 2004

[11] G Maier M Bocciarelli G Bolzon and R Fedele ldquoInverseanalyses in fracture mechanicsrdquo International Journal of Frac-ture vol 138 no 1ndash4 pp 47ndash73 2006

[12] F Yin and D L Gao ldquoMechanical analysis and design of casingin directional well under in-situ stressesrdquo Journal of Natural GasScience and Engineering vol 20 pp 285ndash291 2014

[13] S Sakurai and S Akutagawa ldquoReport some aspects of backanalysis in geotechnical engineeringrdquo in Proceedings of theISRM International Symposium (EUROCK rsquo93) pp 1133ndash1140Lisboa Portugal June 1993

[14] L Z Xue Engineering Optimum Programming TechnologiesTianjing University Press Tianjing China 1998 (Chinese)

[15] F Li J-A Wang and J C Brigham ldquoInverse calculation of insitu stress in rock mass using the Surrogate-Model AcceleratedRandom Search algorithmrdquo Computers and Geotechnics vol 61pp 24ndash32 2014

[16] X Z Wang W Q Liu Y G Sun and Y J Ma ldquoField testingof fracturing techanolgy in coal formed gas wellrdquoOil Drilling ampProduction Technology vol 23 no 2 pp 58ndash61 2001 (Chinese)

[17] J Teng Y B Yao D M Liu and Y D Cai ldquoEvaluation ofcoal texture distributions in the southern Qinshui basin NorthChina investigation by amultiple geophysical loggingmethodrdquoInternational Journal of Coal Geology vol 140 pp 9ndash22 2015

[18] Z Meng J Zhang and R Wang ldquoIn-situ stress pore pressureand stress-dependent permeability in the Southern QinshuiBasinrdquo International Journal of Rock Mechanics and MiningSciences vol 48 no 1 pp 122ndash131 2011

[19] P Lancaster and K Salkauskas ldquoSurfaces generated by movingleast squaresmethodsrdquoMathematics of Computation vol 37 no155 pp 141ndash158 1981

[20] AHMedley ldquoStorage of natural gas in salt cavernsrdquo in Proceed-ings of the SPE Northern Plains Section Regional Meeting SPE-5427-MS Conference Paper Society of Petroleum EngineersOmaha Neb USA May 1975

[21] G Han M S Bruno K Lao J Young and L Dorfmann ldquoGasstorage and operations in single-bedded salt caverns stabilityanalysesrdquo SPEProduction andOperations vol 22 no 3 pp 368ndash376 2007

[22] P Berest and B Brouard ldquoLong-term behavior of salt cavernsrdquoin Proceedings of the 48th US Rock MechanicsGeomechanicsSymposium vol 14 Minneapolis Minn USA June 2014ARMA-2014-7032

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

where119879 = 119863119871119870minus1119879 is the transfermatrix between the nodal

stress matrix and the boundary loads array For the linearelastic body 119879 could be determined when the finite elementmodel of UGS reservoir is established 119865(119883) is uncertainbecause the loads are unknown and only a part of120590 is certainsince the stress values of only a few wells in the reservoir tobe studied are known Therefore in general the boundaryloads parameter vector 119883 cannot be calculated from (8) andis waiting to be discovered

3 Optimisation Method

In situ stress field inversion is used to find the model orparameters that can fit actual data under certain principlesIt is not only a method of data fitting but also a way ofoptimising The inversion result is closely related to theoptimal criterions which is used to evaluate the degree ofdata fitting Sometimes we have prior information aboutsome areas that have already been researched Based on thisprior information we could restrain the optimal criterion andparameters Generally the data going to be used in stressinversion calculation is obtained from the reservoir stressfield and contains the stress of logging fracturing boreholebreakout or core testing of certain wells The principle ofstress inversion (or objective function) could be described asthe followingmodels by using the least squaresmethod as theoptimal criterion

(1) Constraint Model If the stress value of the test well is 120590119871

that is the simulation value of the stress field 120590 from the finiteelement calculation is known the model could be describedas

min Φ (119883) = (120590 minus 120590119871)119879(120590 minus 120590

119871)

st ℎ1le 120593 (119883) le ℎ

2

119886119894le 119883119894le 119887119894

(9)

where 120590 is derived from the finite element calculation Φ(119883)

is the mean squared error between the measurement andcalculation 119883 is the unknown boundary load vector andℎ1and ℎ

2are the boundary load parameter functions of the

upper bound and lower bound respectively determined byprior information 119886

119894and 119887119894are the boundary load parameters

of upper and lower bound respectively

(2) Unconstraint Model An unconstraint model exists whenthe condition in (9) is not met that is

minΦ (119883) = (120590 minus 120590119871)119879(120590 minus 120590

119871) (10)

where symbols are the same as those in (9)

4 Inversion Algorithm

According to the principle of inversion calculation backanalysis is used to calculate the input data of a system byanalysing its output data [13] Therefore if the system is wellconstructed it is possible to obtain the input data from its

output data which is usually achieved through experimentsThe back analysis ensures that a unique solution can begenerated as long as the mechanical model is real and thedata is sufficient

The general way of inversion is to present the parametervector of boundary load 119883 and then calculate the reservoirrsquosstress response value under 119883 During this process thefinite element method is used to approach the actual data ofsome well sites based on the least squares method Usuallythe Gauss-Newton method is adopted to solve the leastsquares problem of (9) or (10) However ill-conditionedcoefficient matrix of the equation low iterative speed andspurious convergence may occur sometimes In this paper adamped least squares method [Levenberg-Marquardt (LM)method] [14] is implemented to approach the parametervector of boundary load 119883 The LM method which has ahigh optimisation speed and precision possesses both theadvantages of the gradient method and the Newton methodWhen the initial value of damping factor 120582

119896is small the step

size is set to be equal to that of the Newton method while itis equal to that of the gradient descent when 120582

119896is large

When the iteration reaches step k the column vector ofstress response and boundary load120590

119896 and119883119896 are setThen120590119896

series expansion around119883119896 is performed and the trace above

the second-order polynomial is ignored to get the followingequation [15]

120590119896= 120590119896minus1

+120597120590

120597119883(119883119896minus 119883119896minus1

)

= 120590119896minus1

+ 119869 sdot (119883119896minus 119883119896minus1

)

(11)

where 119869 is the Jacobi matrix of 120590119896 in119883119896 and it is defined from

the finite difference method as

119869119894119895=

120597120590119894

120597119883119895

asympΔ120590119894

Δ119883119895

=

120590119894(1198831 1198832 119883

119895+ Δ119883119895 119883

119899)

Δ119883119895

minus

120590119894(1198831 1198832 119883

119895 119883

119899)

Δ119883119895

(12)

where 120590119894could be calculated from (7)

The search direction used in LM method is from thesolution of a set of linear equalities The iterative formula is

[119869119879(119883(119896)

) 119869 (119883(119896)

) + 120582119896119868] Δ119883

(119896)

= minus119869119879(119883(119896)

) sdot 120590119896minus1

minus 120590119871

(13)

where Δ119883 is the search direction vector of 119883 and 119868 is theunit matrix Scalar 120582

119896is the damping factor controlling the

direction and value ofΔ119909119896When 120582

119896is 0 the direction ofΔ119909

119896

is the same as the result of the Gauss-Newton method When

4 Mathematical Problems in Engineering

120582119896tends to infinity we could derive the following equation

from (13)

[119869119879(119883(119896)

) 119869 (119883(119896)

) + 120582119896119868] Δ119883

(119896) asymp 120582119896119868 Δ119883

(119896)

= minus119869119879(119883(119896)

) sdot 120590119896minus1

minus 120590119871

(14)

Thus

Δ119883(119896)

= minus1

120582119896

119869119879(119883(119896)

) sdot 120590119896minus1

minus 120590119871 (15)

where Δ119883(119896) is in the direction of the zero vector and has a

steep descending slope for 120582119896 which is large enough that is

Φ(119909(119896)

+ Δ119909(119896)

) lt Φ (119909(119896)

) (16)

From (13) we derived

Δ119883(119896)

= minus [119869119879(119883(119896)

) 119869 (119883(119896)

) + 120582119896119868]minus1

119869119879(119883(119896)

)

sdot 120590119896minus1

minus 120590119871

(17)

Then we got

119883(119896)

= 119883(119896minus1)

+ Δ119883(119896)

(18)

The boundary load vector could be derived from iterativesolving of (13)ndash(18) The key step is how to choose iterativeinitial value If the range of the boundary load is knownwe can set the iterative initial value of 119883 as the average ofthe upper and lower bounds If the range is an unboundeddomain the iterative initial value of 119883 could be determinedafter experiential trials

From the above discussion if the measured data ofreservoir stress is available the boundary load could be deter-mined The load parameter vector and reservoirrsquos horizontalstress 120590

119871could satisfy

119866119883 = 120590119871 (19)

where 119866 isin 119862119898times119899 denotes the transfer matrix 119898 is the point

number for the measurement and 119899 is the dimension of theboundary load vector If 119899 gt 119898 a set of boundary loadvectors could be clear based on the principle of controlling thecomprehensive error to be minimum for the nonuniquenessof the solution

The specific steps of finite element inversion of the in situstress field go as follows Firstly choose a proper objectivefunction Secondly adjust and search parameters by theoptimal method and forward-model the computed value bythe finite element numerical algorithm Thirdly substitutethe measured and calculated values into the objective func-tion and judge the result from the objective function Ifthe result did not reach the minimum the adjusting andsearching should be continued Repeat this process untilthe result from objective function reaches the minimumvalue and finally the boundary load parameter is generatedAfterwards the forward modelling is carried out to acquirethe reservoirrsquos stress field by the boundary load derived

Initial data X

Parameter modification

Inversion parameter and results

Feature analysis of in situ stress field

Forward numerical calculation 120590 = TF(X)

Objective function 120601(X)

Measured data 120590L

120601(X) le 120576

Figure 1 Flowchart of the finite element inversion

Object

P1

P2

P3

P4

P5

P6

P7

P8

P9

P10

P11

P12

Figure 2 Computational model for the boundary load

from the inversion (Figure 1 shows the inversion process)The applied boundary condition is shown in Figure 2 Thecalculation is performed in the entire circle area to avoidstress concentration Assuming the strata are boundless andstress is continuously changed if the stress applied on thecircle can make the inversion results close enough to the testresults the stress applied on the rectangular must be rightThe stress distribution in the object area is the result we wantfrom the inversion

5 Example and Application

Based on the aforementioned theory in order to get theboundary load of the in situ stress field we could programthe finite element iteration by finite element software Inthis paper a Turkey salt cavern is chosen to illustrate theapplication of reversion technology on resolving in situ stressboundary load Furtherly this method is also applied on thein situ stress analysis in Qinshui basin of China [16ndash18] Theleast moving squares method can also be used to deduce

Mathematical Problems in Engineering 5

1150 1100 1150 1150 1200 1200 1250 125012501100

1250

Tuz formation

Cihanbeyli to Kochisar formation

1100

1800

1600

1400

1200

1000

800

600

400

200

0

Dep

th b

elow

surfa

ce (m

)

UG

S12

UG

S11

UG

S10

UG

S1

UG

S2

UG

S9

UG

S8

UG

S7

UG

S6

UG

S5

UG

S4

UG

S3

Figure 3 Section map of UGS in Turkey salt cavern

UGS11

UGS10

UGS1

UGS2UGS9

UGS8

UGS7

UGS6

UGS5 UGS4UGS3

Figure 4 Three-dimensional overview of UGS in Turkey salt cavern

the in situ stress field by carefully selecting the primaryfunction and the weight function [19] The UGS distributionis shown in Figures 3ndash5 Figure 3 shows the well depths ofmultiple UGS wells The minimum depth is 1110m whilethe maximum depth is 1500m Figure 4 shows the three-dimensional distribution of these UGS wells Figure 5 showsthe planar structure of the salt cavern Twelve reservoirs asmany as UGS wells construct the salt cavern All sites are inthe southeast section which is indicated by the blue spotsThe red line in Figure 5 indicates the fault zone of the researcharea

51 Building UGS in a Salt Cavern Salt owns several proper-ties that make it ideal for gas storage [19] It has a moderatelyhigh strength and flows plastically to close fracture that couldotherwise become a leak The porosity and permeability ofsalt to liquid and gaseous hydrocarbons are close to zerothereby preventing stored gas escape [20 21] Salt caverns canprovide high deliverability and gas can be withdrawn quicklybecause no pressure from flow is lost through the pores Asalt cavern in south Turkey is used to construct undergroundnatural gas storage to compensate season-related fluctuations

of gas consumption Moreover by a salt cavern gas can beprovided to customers under optimum economic conditionsbetween buying and selling To construct UGS by a saltcavern a well should be drilled first and then water shouldbe injected to dissolve the salt Brine is pumped out of the saltcavern and a cavity is formed which can be treated to storenatural gas

52 Calculation Model and Boundary Conditions Based onthe prerequisite of the numerical simulation a geologicmodel the target layer isolation body of a rock block is takenas the simulation object Two rules should be followed todetermine the computational area (1) extend the geometricrange to eliminate or weaken the effects from the bound-ary conditions on the research area and (2) the geometryconstraint conditions on the boundary are easy to get Ingeneral the layered distribution of underground sedimentaryrock could be treated as a plane problem of elastic mechanicsin the comprehensive mathematical model Figure 5 is themechanical model of the research area which is a rectangulararea with a length of 8 km and width of 32 km Three majorfault zones are taken into consideration The lengths and

6 Mathematical Problems in Engineering

UGS11UGS12

UGS1UGS10

UGS2 UGS9UGS8 UGS7

UGS6 UGS5UGS4 UGS3

0 1000 2000(m)

Figure 5 Planar structure of the research area and the distributionof the UGS

widths of fault zones are 8 km lowast 1 km 12 km lowast 08 km and35 km lowast 08 km respectively

The following eight factors are taken as the basic factorsof tectonic force compressive or tensile forces and shearforces in the east west south and north The boundaryload is adjusted depending on the size of research area Asthe stress field is continuously changed the external stressin the circle can be divided into 12 parts which is largeenough to guarantee the accuracy of calculation By changingthe value of the 12 boundary forces we could simulate thecompressive or tensile forces in the direction of eastndashwest andsouthndashnorth as well as shear deformation in the horizontalplane to get the in situ stress field distribution of the partialresearch area At the same time boundary force constraintis applied to make the value of the resultant force and theresultant moment zero (Figure 6 shows the method to applythe boundary force)The edge points of the lower boundary ofthemodel are fixed to ensure no rigid translation and rotationin the research area

Generally a geologic body is taken as a rock with auniform block to confirm the rock physical parameters fromdifferent tectonic positions The basic assumption is thateach fault zone is homogeneous The rock strength Youngrsquosmodulus and other rock mechanical parameters should bedecreased according to the rock properties in specific depth(Table 1 shows information about mechanical parameterschosen by the model) The cored sample is taken from the

1ElementsMAT Num

Figure 6 Mechanical model of the calculated area

Table 1 Rock mechanics parameters

Number Media type Elasticity modulusMPa Poissonrsquos ratio ]1 Reservoir 31200 02252 Fault zone 1 22500 02703 Fault zone 2 23500 02604 Fault zone 3 24000 02905 Other layers 35200 0220

spot field and by indoor experiment elasticity modulus andPoissonrsquos ratio are obtained The grids of the model adoptedmainly are four-node isoperimetric rectangular elements andfor part of the area triangle elements are used In additionthe element length of the fault zone and its surrounding areais shorter and the grids are relatively finer than those of otherparts

53 Inversion of the In Situ Stress Field of the Salt Cavern

(1) Initial boundary conditions the initial values of thedistributed normal stress 119875

1sim 11987512in the east south

west and north could be set as an arbitrary value with10MPa here (Figure 6 shows how the boundary forceacts)

(2) Constraint object based on the reference about insitu stress measurement we chose 10 values of in situstress and five directions from eight survey points asknown constraint parameters to inverse 12 unknownboundary forces (Figure 5 shows the distribution ofconstraint points and measured data are shown inTables 2 and 3)

(3) Searching range the range of the boundary nor-mal stress constraint value is 1MPasim30MPa (lowerboundsimupper bound) and in the same way therange of the boundary shear stress constraint valueminus10MPasim10MPa The precision of the iteration ter-mination is 10minus5MPa2The boundary force constraintshould be taken into consideration to make sure thatthe resultant moment of the boundary force is zero

Mathematical Problems in Engineering 7

Table 2 Comparison between the inversion results and the measured values on principal stress magnitude

Measured points Stress components Inversion valueMPa Measured valueMPa Relative error

UGS6 120590119867

minus2169 minus2323 66120590ℎ

minus2793 minus2777 058

UGS7 120590119867

minus2132 minus1996 681120590ℎ

minus2602 minus261 031

UGS8 120590119867

minus2121 minus2026 469120590ℎ

minus2725 minus2852 445

UGS9 120590119867

minus2066 minus2226 719120590ℎ

minus2783 minus2664 447

UGS10 120590119867

minus1906 minus1856 269120590ℎ

minus2515 minus2385 545

Table 3 Comparison between the inverted results and themeasuredvalues on the direction of maximum principal stress

Measuredpoints Inversion value(∘) Measured value(∘) Error(∘)

UGS6 NE57327 NE55 2327UGS7 NE51798 NE535 minus1702

UGS8 NE52334 NE549 minus257

UGS9 NE4132 NE451 minus378

UGS10 NE51713 NE536 minus1887

The relevant constrained inversion model could bederived from (9)

(4) Inversion result based on the conditions above thecalculation results on the survey points t from inver-sion are shown in Tables 2 and 3 where 120590

119867and

120590ℎare the maximum and minimum principal stress

magnitude respectively The measurement resultsfrom well logging are also presented and comparedwith the inversion results Table 4 shows the results ofthe boundary load inversion

Tables 2 and 3 show that the maximum relative errorsof stress components 120590

119867and 120590

ℎare controlled within 8

The errors of the direction of maximum principal stress arecontrolled within 4∘ All the stress values obtained from theinversion reached or are close to the actual data and theprecision of the inversion results could meet engineeringrequirements

We applied the optimised boundary force from the inver-sion to perform forward calculation This approach couldlead to the distribution of the stress field value and directionof the Turkey salt cavern (shown in Figures 7 and 8)

6 Stress Distribution Character of the Wellnear the Hole of the UGS

On the basis of the analysis above the two principal stresses120590119867and 120590ℎof the 12 wells are determined Given that the stress

concentration occurs by the hole which will influence thestability of the shaft wall the von Mises stress distributioncharacter near the shaft wall (0ndash255m) should be checked

minus29

minus13minus14

minus15

minus16

minus17

minus18

minus19

minus20

minus21

minus22

minus23

minus24

minus25

minus26

minus27

minus28

minus29

minus30

minus31

minus32

minus33

minus34

minus24

minus19

minus19

minus19

minus19

minus19minus19

2000

4000

6000

8000

10000

12000

14000

4000 6000 8000 100002000

Figure 7 Distribution of the maximum principal stress and orien-tation

up as shown in Figure 9 Here well UGS1 is taken as anexample with well diameter of UGS1 279 cm bottom holepressure 12MPa overburden pressure 231MPa and welldepth 1100m

As shown in Figure 9 given that stress nonhomogeneouswith the in situ stress exists the von Mises stress distributionof the shaft wall is in an oval shape The von Mises stresshas a higher value in the shaft wall which is within 281ndash568MPa The von Mises stress decreases rapidly away fromthe shaft wall and the shape changes to a butterfly-like oneThe maximum von Mises stress decreases to 171MPa when06m away from the shaftwall It is worthmentioning that theresult here contains the influence from stress concentrationaround the shaft wall As shown in Figures 9(d) and 9(e) thevonMises stress owns the samedistribution pattern and tendsto be steady far from the shaft wall

8 Mathematical Problems in Engineering

Table 4 Optimum results of the boundary load unit MPa

1198751

1198752

1198753

1198754

1198755

1198756

1198757

1198758

1198759

11987510

11987511

11987512

2639 1999 75277 2113 2275 9216 5328 12053 1484 1021 5257 2977

2000

4000

6000

8000

10000

12000

14000

4000 6000 8000 100002000

minus24

minus38

minus43

minus43

minus43

minus38

minus43

minus43

minus33

minus38

minus28

minus33

minus38minus43

minus25

minus26

minus27

minus28

minus29

minus30

minus31

minus32

minus33

minus34

minus35

minus36

minus37

minus38

minus39

minus40

minus41

minus42

minus43

minus44

minus45

minus46

minus47

minus48

Figure 8 Distribution of the minimum principal stress and orien-tation

As shown in Figure 10 just like the von Mises stress theshearing stress distribution is in an oval shape near the shaftwall and changes to a butterfly-like shape from the shaft wallThe shearing stress on the shaft wall is maximum (291MPa)due to the stress concentration and it decreases rapidly awayfrom the shaft wall with 89MPa when 06m away Shearingstress tends to be stable when it is 16m away from the shaftwall

From the stress analysis above we can draw the conclu-sion as the following

The stress concentration phenomenon does exist aroundthe hole The von Mises stress and shearing stress aremaximum on the shaft wall The stress distribution is in anoval shape around the shaft wall while it is in a butterfly-likeshape near the hole Both stresses decrease rapidly when theyare 1m away from the shaft wall and tend to be stable when13m away from the shaft wall

7 The Hole Stability Analysis of UGS

71 The Rock Fracture Analysis around the Hole of UGSBased on the Griffith Principle Griffith defined the inherentdefects or micro-fracture in the material as the Griffithfracture Based on theGriffith criterion [22] the rock fracturecondition in the planar stress state is shown below

When (1205901+ 31205902) gt 0 the fracture criterion is

(1205902minus 1205901)2= 8 (120590

2+ 1205901) 120590119905 (20)

Equation (20) can also be like 120590119905= (1205902minus 1205901)28(1205902+ 1205901)

cos 2120573 = (1205901minus 1205902)2(1205902+ 1205901)

When (1205901+ 31205902) le 0 the fracture criterion is

1205902= minus120590119905

sin 2120573 = 0

(21)

where 120590119905is the tensile stress from the rock unidirectional

tensile test 120573 is the angle between the long-axis of theoval fracture and the principal compressive stress axis Thefracture angle is 120579 = 120573

If a new fracture occurs it will be in the direction of ovalfractureThe angle between the new fracture and the originalone is 120595 (Figure 11) The following equation can be obtainedby the fracture criterion

120595 = minus2120573 (22)

Equation (22) shows that the angle between the newfracture and the original one is 2120573 in the condition of (120590

1+

31205902) gt 0 The negative sign indicates the direction of the

new fracture is clockwise rotation by 120573 from the principalcompression stress axis (Figure 11)

For the three-dimensional problem Griffith fracturecriterion can be depicted as follows

When (1205901+ 31205903) gt 0 the fracture criterion is

(1205901minus 1205902)2+ (1205902minus 1205903)2+ (1205903minus 1205901)2

= 24120590119905(1205901+ 1205902+ 1205903)

cos 2120579 =1205901minus 1205903

2 (1205901+ 1205903)

(23)

When (1205901+ 31205903) le 0 the fracture criterion is

120590119905= minus1205903 (24)

Based on the fracture criterion above it can be judgedonly whether the tensile fracture or the shear fracture occurswhile the developing degrees of the fracture cannot bedetermined In fact with the asymmetry of the tectonic stressdistribution the fracture developing degrees of tectonic partdiffer The concept of fracture factor is introduced to judgethe fracture developing degrees of the rock quantitativelyThefracture factor is given by

119868120590=

100381610038161003816100381610038161003816100381610038161003816

120590119905

[120590119905]

100381610038161003816100381610038161003816100381610038161003816

or 119868119905=

100381610038161003816100381610038161003816100381610038161003816

120591119899

[120591119899]

100381610038161003816100381610038161003816100381610038161003816

(25)

Mathematical Problems in Engineering 9

568

568

281281

(a) 000m around the shaft wall

171

171

4343

(b) 060m around the shaft wall104

104

3737

(c) 105m around the shaft wall

85

85

5858

(d) 165m around the shaft wall

78

78

6767

(e) 255m around the shaft wall

Figure 9 von Mises stress distribution of the shaft wall in well UGS1

10 Mathematical Problems in Engineering

291

291

125125

(a) 00m around the shaft wall

89

89

1515

(b) 060m around the shaft wall

57

57

2323

(c) 105m around the shaft wall

47

47

3434

(d) 165m around the shaft wall

44

44

3838

(e) 255m around the shaft wall

Figure 10 The maximum shearing stress distribution around the shaft wall in UGS1

Mathematical Problems in Engineering 11

Table 5 Hole rock tensile fracture analysis results in UGS

Well Maximum principalstressMPa

VerticalstressMPa

Minimum principalstressMPa

PorepressureMPa

TensilestressMPa

Tensile strength[120590119905]MPa

Tensile fracturefactor 119868

120590

UGS6 minus2169 minus2597 minus2793 1200 095 497 0191UGS7 minus2132 minus2610 minus2602 1200 093 484 0192UGS8 minus2121 minus2503 minus2725 1200 099 492 0201UGS9 minus2066 minus2310 minus2783 1200 092 494 0186UGS10 minus1906 minus2515 minus2215 1200 098 495 0198

Table 6 Hole rock shear fracture analysis results in UGS

Well Maximum principalstressMPa

VerticalstressMPa

Minimum principalstressMPa

CohesivestressMPa

Maximum shearingstressMPa

Shear strength[120591119899]MPa

Shear fracturefactor 119868

119905

UGS6 minus2169 minus2597 minus2793 466 249 354 0703UGS7 minus2132 minus2610 minus2602 435 222 368 0603UGS8 minus2121 minus2503 minus2725 405 227 372 0510UGS9 minus2066 minus2310 minus2783 462 213 347 0614UGS10 minus1906 minus2515 minus2215 450 269 357 0754

The newfracture

The newfracture

x

y

120595 = minus2120573

120595 = minus2120573

120573

12059011205901

1205902

1205902

Figure 11 Fracture expansion direction of the oval fracture

where 119868120590and 119868119905are tensile fracture factor and shear fracture

factor 120590119905and 120591

119899are tensile stress and shearing stress in

the fracture plane [120590119905] and [120591

119899] are tensile strength and

shearing resistance The shaft rock will not be cracked whenthe fracture factor is below 1

72 Rock Fracture Analysis Results in the UGS According tothe Griffith fracture criterion presented above we conductthe tensile fracture analysis on the rock of the sidewall inUGS of a Turkey salt cavern With the obtained maximumand minimum horizontal in situ stresses of the test wells

r

120590h1

120590h2

120590h2

120590h1

120590120579120591r120579

120590r

Figure 12 The calculation model of wellbore stress

the tensile stress and shearing stress can be calculated bythe classical elastic theory The calculation model is shownin Figure 12 where 120590

ℎ1 120590ℎ2are the maximum and minimum

horizontal principal in situ stresses 120590119903is the radial normal

stress 120590120579is the tangential normal stress and 120591

119903120579is the

shearing stressThe results are shown in Table 5 showing that the

maximum tensile stress is smaller than the tensile strengthTherefore on the rock of the shaft wall no fracture will bedeveloped

Moreover according to the Coulomb-Navier failure cri-terion the shear fracture analysis is conducted on the rock ofthe shaft wall in UGS of a Turkey salt cavern Table 6 showsthe analysis results The shear fracture factors 119868

119905 in the hole

are all under 1 Given that themaximum shearing stress is lessthan the shearing strength the rock of the shaft wall will notdevelop a shear fracture

12 Mathematical Problems in Engineering

8 Conclusion

In this paper we improved the nonuniqueness of the in situstress field inversion solution by a priori constraints Fromthe perspective of basic equations of elastic mechanics anin situ stress numerical inversion model is built and basicfactors of in situ stress field are considered in applying thefinite element inversion method on the boundary loads ofthe in situ stress field An appropriate geometrical modeland suitable boundary conditions ensured an accurate in situstress field is derived The method is successfully applied inUGS of a Turkey salt cavern The optimisation results onprincipal stress are close to those from measurement By theoptimisation the overall in situ stress field of the researcharea is obtained and the stress distribution of the shaft wallis presented according to the calculation results The safetyof the shaft wall is confirmed by applying the correspondingfracture criterion

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to acknowledge the financial support ofthe National Natural Science Foundation of China (Grantsnos 51374228 51274231 and U1262208) and the CNPC KeyLaboratory Fund (Contract no 2014A-4214)

References

[1] B S Aadnoy ldquoInversion technique to determine the in-situstress field from fracturing datardquo Journal of Petroleum Scienceand Engineering vol 4 no 2 pp 127ndash141 1990

[2] M Bloch C A M Siqueira F H Ferreira and J C JConceicao ldquoTechniques for determining in-situ stress directionand magnitudesrdquo in Proceedings of the Latin American andCaribbean Petroleum Engineering Conference SPE 39075-MSRio de Janeiro Brazil August-September 1997

[3] J Huang D V Griffiths and S-W Wong ldquoIn situ stress deter-mination from inversion of hydraulic fracturing datardquo Interna-tional Journal of Rock Mechanics and Mining Sciences vol 48no 3 pp 476ndash481 2011

[4] J Djurhuus and B S Aadnoslashy ldquoIn situ stress state from inversionof fracturing data from oil wells and borehole image logsrdquoJournal of Petroleum Science and Engineering vol 38 no 3-4pp 121ndash130 2003

[5] B S Aadnoy ldquoIn-situ stress directions from borehole fracturetracesrdquo Journal of Petroleum Science and Engineering vol 4 no2 pp 143ndash153 1990

[6] C A Barton D Moos P Peska and M D Zoback ldquoUtilizingwellbore image data to determine the complete stress tensorapplication to permeability anisotropy and wellbore stabilityrdquoThe Log Analyst vol 38 no 6 pp 21ndash33 1997

[7] H Zhao F Ma J Xu and J Guo ldquoIn situ stress field inversionand its application in mining-induced rock mass movementrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 53 pp 120ndash128 2012

[8] Z D Chen Q A Meng and T F Wan ldquoNumerical simulationof tectonic stress field in Gulong depression in Songliao Basinusing elasitic-plastic increment methodrdquo Earth Science Fron-tiers vol 9 no 2 pp 483ndash491 2002 (Chinese)

[9] D K Singha and R Chatterjee ldquoGeomechanical modelingusing finite element method for prediction of in-situ stress inKrishna-Godavari basin Indiardquo International Journal of RockMechanics and Mining Sciences vol 73 no 5 pp 15ndash27 2015

[10] G Maier R Ardito and R Fedele ldquoInverse analysis problemsin structural engineering of concrete damsrdquo ComputationalMechanics vol 9 pp 5ndash10 2004

[11] G Maier M Bocciarelli G Bolzon and R Fedele ldquoInverseanalyses in fracture mechanicsrdquo International Journal of Frac-ture vol 138 no 1ndash4 pp 47ndash73 2006

[12] F Yin and D L Gao ldquoMechanical analysis and design of casingin directional well under in-situ stressesrdquo Journal of Natural GasScience and Engineering vol 20 pp 285ndash291 2014

[13] S Sakurai and S Akutagawa ldquoReport some aspects of backanalysis in geotechnical engineeringrdquo in Proceedings of theISRM International Symposium (EUROCK rsquo93) pp 1133ndash1140Lisboa Portugal June 1993

[14] L Z Xue Engineering Optimum Programming TechnologiesTianjing University Press Tianjing China 1998 (Chinese)

[15] F Li J-A Wang and J C Brigham ldquoInverse calculation of insitu stress in rock mass using the Surrogate-Model AcceleratedRandom Search algorithmrdquo Computers and Geotechnics vol 61pp 24ndash32 2014

[16] X Z Wang W Q Liu Y G Sun and Y J Ma ldquoField testingof fracturing techanolgy in coal formed gas wellrdquoOil Drilling ampProduction Technology vol 23 no 2 pp 58ndash61 2001 (Chinese)

[17] J Teng Y B Yao D M Liu and Y D Cai ldquoEvaluation ofcoal texture distributions in the southern Qinshui basin NorthChina investigation by amultiple geophysical loggingmethodrdquoInternational Journal of Coal Geology vol 140 pp 9ndash22 2015

[18] Z Meng J Zhang and R Wang ldquoIn-situ stress pore pressureand stress-dependent permeability in the Southern QinshuiBasinrdquo International Journal of Rock Mechanics and MiningSciences vol 48 no 1 pp 122ndash131 2011

[19] P Lancaster and K Salkauskas ldquoSurfaces generated by movingleast squaresmethodsrdquoMathematics of Computation vol 37 no155 pp 141ndash158 1981

[20] AHMedley ldquoStorage of natural gas in salt cavernsrdquo in Proceed-ings of the SPE Northern Plains Section Regional Meeting SPE-5427-MS Conference Paper Society of Petroleum EngineersOmaha Neb USA May 1975

[21] G Han M S Bruno K Lao J Young and L Dorfmann ldquoGasstorage and operations in single-bedded salt caverns stabilityanalysesrdquo SPEProduction andOperations vol 22 no 3 pp 368ndash376 2007

[22] P Berest and B Brouard ldquoLong-term behavior of salt cavernsrdquoin Proceedings of the 48th US Rock MechanicsGeomechanicsSymposium vol 14 Minneapolis Minn USA June 2014ARMA-2014-7032

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

120582119896tends to infinity we could derive the following equation

from (13)

[119869119879(119883(119896)

) 119869 (119883(119896)

) + 120582119896119868] Δ119883

(119896) asymp 120582119896119868 Δ119883

(119896)

= minus119869119879(119883(119896)

) sdot 120590119896minus1

minus 120590119871

(14)

Thus

Δ119883(119896)

= minus1

120582119896

119869119879(119883(119896)

) sdot 120590119896minus1

minus 120590119871 (15)

where Δ119883(119896) is in the direction of the zero vector and has a

steep descending slope for 120582119896 which is large enough that is

Φ(119909(119896)

+ Δ119909(119896)

) lt Φ (119909(119896)

) (16)

From (13) we derived

Δ119883(119896)

= minus [119869119879(119883(119896)

) 119869 (119883(119896)

) + 120582119896119868]minus1

119869119879(119883(119896)

)

sdot 120590119896minus1

minus 120590119871

(17)

Then we got

119883(119896)

= 119883(119896minus1)

+ Δ119883(119896)

(18)

The boundary load vector could be derived from iterativesolving of (13)ndash(18) The key step is how to choose iterativeinitial value If the range of the boundary load is knownwe can set the iterative initial value of 119883 as the average ofthe upper and lower bounds If the range is an unboundeddomain the iterative initial value of 119883 could be determinedafter experiential trials

From the above discussion if the measured data ofreservoir stress is available the boundary load could be deter-mined The load parameter vector and reservoirrsquos horizontalstress 120590

119871could satisfy

119866119883 = 120590119871 (19)

where 119866 isin 119862119898times119899 denotes the transfer matrix 119898 is the point

number for the measurement and 119899 is the dimension of theboundary load vector If 119899 gt 119898 a set of boundary loadvectors could be clear based on the principle of controlling thecomprehensive error to be minimum for the nonuniquenessof the solution

The specific steps of finite element inversion of the in situstress field go as follows Firstly choose a proper objectivefunction Secondly adjust and search parameters by theoptimal method and forward-model the computed value bythe finite element numerical algorithm Thirdly substitutethe measured and calculated values into the objective func-tion and judge the result from the objective function Ifthe result did not reach the minimum the adjusting andsearching should be continued Repeat this process untilthe result from objective function reaches the minimumvalue and finally the boundary load parameter is generatedAfterwards the forward modelling is carried out to acquirethe reservoirrsquos stress field by the boundary load derived

Initial data X

Parameter modification

Inversion parameter and results

Feature analysis of in situ stress field

Forward numerical calculation 120590 = TF(X)

Objective function 120601(X)

Measured data 120590L

120601(X) le 120576

Figure 1 Flowchart of the finite element inversion

Object

P1

P2

P3

P4

P5

P6

P7

P8

P9

P10

P11

P12

Figure 2 Computational model for the boundary load

from the inversion (Figure 1 shows the inversion process)The applied boundary condition is shown in Figure 2 Thecalculation is performed in the entire circle area to avoidstress concentration Assuming the strata are boundless andstress is continuously changed if the stress applied on thecircle can make the inversion results close enough to the testresults the stress applied on the rectangular must be rightThe stress distribution in the object area is the result we wantfrom the inversion

5 Example and Application

Based on the aforementioned theory in order to get theboundary load of the in situ stress field we could programthe finite element iteration by finite element software Inthis paper a Turkey salt cavern is chosen to illustrate theapplication of reversion technology on resolving in situ stressboundary load Furtherly this method is also applied on thein situ stress analysis in Qinshui basin of China [16ndash18] Theleast moving squares method can also be used to deduce

Mathematical Problems in Engineering 5

1150 1100 1150 1150 1200 1200 1250 125012501100

1250

Tuz formation

Cihanbeyli to Kochisar formation

1100

1800

1600

1400

1200

1000

800

600

400

200

0

Dep

th b

elow

surfa

ce (m

)

UG

S12

UG

S11

UG

S10

UG

S1

UG

S2

UG

S9

UG

S8

UG

S7

UG

S6

UG

S5

UG

S4

UG

S3

Figure 3 Section map of UGS in Turkey salt cavern

UGS11

UGS10

UGS1

UGS2UGS9

UGS8

UGS7

UGS6

UGS5 UGS4UGS3

Figure 4 Three-dimensional overview of UGS in Turkey salt cavern

the in situ stress field by carefully selecting the primaryfunction and the weight function [19] The UGS distributionis shown in Figures 3ndash5 Figure 3 shows the well depths ofmultiple UGS wells The minimum depth is 1110m whilethe maximum depth is 1500m Figure 4 shows the three-dimensional distribution of these UGS wells Figure 5 showsthe planar structure of the salt cavern Twelve reservoirs asmany as UGS wells construct the salt cavern All sites are inthe southeast section which is indicated by the blue spotsThe red line in Figure 5 indicates the fault zone of the researcharea

51 Building UGS in a Salt Cavern Salt owns several proper-ties that make it ideal for gas storage [19] It has a moderatelyhigh strength and flows plastically to close fracture that couldotherwise become a leak The porosity and permeability ofsalt to liquid and gaseous hydrocarbons are close to zerothereby preventing stored gas escape [20 21] Salt caverns canprovide high deliverability and gas can be withdrawn quicklybecause no pressure from flow is lost through the pores Asalt cavern in south Turkey is used to construct undergroundnatural gas storage to compensate season-related fluctuations

of gas consumption Moreover by a salt cavern gas can beprovided to customers under optimum economic conditionsbetween buying and selling To construct UGS by a saltcavern a well should be drilled first and then water shouldbe injected to dissolve the salt Brine is pumped out of the saltcavern and a cavity is formed which can be treated to storenatural gas

52 Calculation Model and Boundary Conditions Based onthe prerequisite of the numerical simulation a geologicmodel the target layer isolation body of a rock block is takenas the simulation object Two rules should be followed todetermine the computational area (1) extend the geometricrange to eliminate or weaken the effects from the bound-ary conditions on the research area and (2) the geometryconstraint conditions on the boundary are easy to get Ingeneral the layered distribution of underground sedimentaryrock could be treated as a plane problem of elastic mechanicsin the comprehensive mathematical model Figure 5 is themechanical model of the research area which is a rectangulararea with a length of 8 km and width of 32 km Three majorfault zones are taken into consideration The lengths and

6 Mathematical Problems in Engineering

UGS11UGS12

UGS1UGS10

UGS2 UGS9UGS8 UGS7

UGS6 UGS5UGS4 UGS3

0 1000 2000(m)

Figure 5 Planar structure of the research area and the distributionof the UGS

widths of fault zones are 8 km lowast 1 km 12 km lowast 08 km and35 km lowast 08 km respectively

The following eight factors are taken as the basic factorsof tectonic force compressive or tensile forces and shearforces in the east west south and north The boundaryload is adjusted depending on the size of research area Asthe stress field is continuously changed the external stressin the circle can be divided into 12 parts which is largeenough to guarantee the accuracy of calculation By changingthe value of the 12 boundary forces we could simulate thecompressive or tensile forces in the direction of eastndashwest andsouthndashnorth as well as shear deformation in the horizontalplane to get the in situ stress field distribution of the partialresearch area At the same time boundary force constraintis applied to make the value of the resultant force and theresultant moment zero (Figure 6 shows the method to applythe boundary force)The edge points of the lower boundary ofthemodel are fixed to ensure no rigid translation and rotationin the research area

Generally a geologic body is taken as a rock with auniform block to confirm the rock physical parameters fromdifferent tectonic positions The basic assumption is thateach fault zone is homogeneous The rock strength Youngrsquosmodulus and other rock mechanical parameters should bedecreased according to the rock properties in specific depth(Table 1 shows information about mechanical parameterschosen by the model) The cored sample is taken from the

1ElementsMAT Num

Figure 6 Mechanical model of the calculated area

Table 1 Rock mechanics parameters

Number Media type Elasticity modulusMPa Poissonrsquos ratio ]1 Reservoir 31200 02252 Fault zone 1 22500 02703 Fault zone 2 23500 02604 Fault zone 3 24000 02905 Other layers 35200 0220

spot field and by indoor experiment elasticity modulus andPoissonrsquos ratio are obtained The grids of the model adoptedmainly are four-node isoperimetric rectangular elements andfor part of the area triangle elements are used In additionthe element length of the fault zone and its surrounding areais shorter and the grids are relatively finer than those of otherparts

53 Inversion of the In Situ Stress Field of the Salt Cavern

(1) Initial boundary conditions the initial values of thedistributed normal stress 119875

1sim 11987512in the east south

west and north could be set as an arbitrary value with10MPa here (Figure 6 shows how the boundary forceacts)

(2) Constraint object based on the reference about insitu stress measurement we chose 10 values of in situstress and five directions from eight survey points asknown constraint parameters to inverse 12 unknownboundary forces (Figure 5 shows the distribution ofconstraint points and measured data are shown inTables 2 and 3)

(3) Searching range the range of the boundary nor-mal stress constraint value is 1MPasim30MPa (lowerboundsimupper bound) and in the same way therange of the boundary shear stress constraint valueminus10MPasim10MPa The precision of the iteration ter-mination is 10minus5MPa2The boundary force constraintshould be taken into consideration to make sure thatthe resultant moment of the boundary force is zero

Mathematical Problems in Engineering 7

Table 2 Comparison between the inversion results and the measured values on principal stress magnitude

Measured points Stress components Inversion valueMPa Measured valueMPa Relative error

UGS6 120590119867

minus2169 minus2323 66120590ℎ

minus2793 minus2777 058

UGS7 120590119867

minus2132 minus1996 681120590ℎ

minus2602 minus261 031

UGS8 120590119867

minus2121 minus2026 469120590ℎ

minus2725 minus2852 445

UGS9 120590119867

minus2066 minus2226 719120590ℎ

minus2783 minus2664 447

UGS10 120590119867

minus1906 minus1856 269120590ℎ

minus2515 minus2385 545

Table 3 Comparison between the inverted results and themeasuredvalues on the direction of maximum principal stress

Measuredpoints Inversion value(∘) Measured value(∘) Error(∘)

UGS6 NE57327 NE55 2327UGS7 NE51798 NE535 minus1702

UGS8 NE52334 NE549 minus257

UGS9 NE4132 NE451 minus378

UGS10 NE51713 NE536 minus1887

The relevant constrained inversion model could bederived from (9)

(4) Inversion result based on the conditions above thecalculation results on the survey points t from inver-sion are shown in Tables 2 and 3 where 120590

119867and

120590ℎare the maximum and minimum principal stress

magnitude respectively The measurement resultsfrom well logging are also presented and comparedwith the inversion results Table 4 shows the results ofthe boundary load inversion

Tables 2 and 3 show that the maximum relative errorsof stress components 120590

119867and 120590

ℎare controlled within 8

The errors of the direction of maximum principal stress arecontrolled within 4∘ All the stress values obtained from theinversion reached or are close to the actual data and theprecision of the inversion results could meet engineeringrequirements

We applied the optimised boundary force from the inver-sion to perform forward calculation This approach couldlead to the distribution of the stress field value and directionof the Turkey salt cavern (shown in Figures 7 and 8)

6 Stress Distribution Character of the Wellnear the Hole of the UGS

On the basis of the analysis above the two principal stresses120590119867and 120590ℎof the 12 wells are determined Given that the stress

concentration occurs by the hole which will influence thestability of the shaft wall the von Mises stress distributioncharacter near the shaft wall (0ndash255m) should be checked

minus29

minus13minus14

minus15

minus16

minus17

minus18

minus19

minus20

minus21

minus22

minus23

minus24

minus25

minus26

minus27

minus28

minus29

minus30

minus31

minus32

minus33

minus34

minus24

minus19

minus19

minus19

minus19

minus19minus19

2000

4000

6000

8000

10000

12000

14000

4000 6000 8000 100002000

Figure 7 Distribution of the maximum principal stress and orien-tation

up as shown in Figure 9 Here well UGS1 is taken as anexample with well diameter of UGS1 279 cm bottom holepressure 12MPa overburden pressure 231MPa and welldepth 1100m

As shown in Figure 9 given that stress nonhomogeneouswith the in situ stress exists the von Mises stress distributionof the shaft wall is in an oval shape The von Mises stresshas a higher value in the shaft wall which is within 281ndash568MPa The von Mises stress decreases rapidly away fromthe shaft wall and the shape changes to a butterfly-like oneThe maximum von Mises stress decreases to 171MPa when06m away from the shaftwall It is worthmentioning that theresult here contains the influence from stress concentrationaround the shaft wall As shown in Figures 9(d) and 9(e) thevonMises stress owns the samedistribution pattern and tendsto be steady far from the shaft wall

8 Mathematical Problems in Engineering

Table 4 Optimum results of the boundary load unit MPa

1198751

1198752

1198753

1198754

1198755

1198756

1198757

1198758

1198759

11987510

11987511

11987512

2639 1999 75277 2113 2275 9216 5328 12053 1484 1021 5257 2977

2000

4000

6000

8000

10000

12000

14000

4000 6000 8000 100002000

minus24

minus38

minus43

minus43

minus43

minus38

minus43

minus43

minus33

minus38

minus28

minus33

minus38minus43

minus25

minus26

minus27

minus28

minus29

minus30

minus31

minus32

minus33

minus34

minus35

minus36

minus37

minus38

minus39

minus40

minus41

minus42

minus43

minus44

minus45

minus46

minus47

minus48

Figure 8 Distribution of the minimum principal stress and orien-tation

As shown in Figure 10 just like the von Mises stress theshearing stress distribution is in an oval shape near the shaftwall and changes to a butterfly-like shape from the shaft wallThe shearing stress on the shaft wall is maximum (291MPa)due to the stress concentration and it decreases rapidly awayfrom the shaft wall with 89MPa when 06m away Shearingstress tends to be stable when it is 16m away from the shaftwall

From the stress analysis above we can draw the conclu-sion as the following

The stress concentration phenomenon does exist aroundthe hole The von Mises stress and shearing stress aremaximum on the shaft wall The stress distribution is in anoval shape around the shaft wall while it is in a butterfly-likeshape near the hole Both stresses decrease rapidly when theyare 1m away from the shaft wall and tend to be stable when13m away from the shaft wall

7 The Hole Stability Analysis of UGS

71 The Rock Fracture Analysis around the Hole of UGSBased on the Griffith Principle Griffith defined the inherentdefects or micro-fracture in the material as the Griffithfracture Based on theGriffith criterion [22] the rock fracturecondition in the planar stress state is shown below

When (1205901+ 31205902) gt 0 the fracture criterion is

(1205902minus 1205901)2= 8 (120590

2+ 1205901) 120590119905 (20)

Equation (20) can also be like 120590119905= (1205902minus 1205901)28(1205902+ 1205901)

cos 2120573 = (1205901minus 1205902)2(1205902+ 1205901)

When (1205901+ 31205902) le 0 the fracture criterion is

1205902= minus120590119905

sin 2120573 = 0

(21)

where 120590119905is the tensile stress from the rock unidirectional

tensile test 120573 is the angle between the long-axis of theoval fracture and the principal compressive stress axis Thefracture angle is 120579 = 120573

If a new fracture occurs it will be in the direction of ovalfractureThe angle between the new fracture and the originalone is 120595 (Figure 11) The following equation can be obtainedby the fracture criterion

120595 = minus2120573 (22)

Equation (22) shows that the angle between the newfracture and the original one is 2120573 in the condition of (120590

1+

31205902) gt 0 The negative sign indicates the direction of the

new fracture is clockwise rotation by 120573 from the principalcompression stress axis (Figure 11)

For the three-dimensional problem Griffith fracturecriterion can be depicted as follows

When (1205901+ 31205903) gt 0 the fracture criterion is

(1205901minus 1205902)2+ (1205902minus 1205903)2+ (1205903minus 1205901)2

= 24120590119905(1205901+ 1205902+ 1205903)

cos 2120579 =1205901minus 1205903

2 (1205901+ 1205903)

(23)

When (1205901+ 31205903) le 0 the fracture criterion is

120590119905= minus1205903 (24)

Based on the fracture criterion above it can be judgedonly whether the tensile fracture or the shear fracture occurswhile the developing degrees of the fracture cannot bedetermined In fact with the asymmetry of the tectonic stressdistribution the fracture developing degrees of tectonic partdiffer The concept of fracture factor is introduced to judgethe fracture developing degrees of the rock quantitativelyThefracture factor is given by

119868120590=

100381610038161003816100381610038161003816100381610038161003816

120590119905

[120590119905]

100381610038161003816100381610038161003816100381610038161003816

or 119868119905=

100381610038161003816100381610038161003816100381610038161003816

120591119899

[120591119899]

100381610038161003816100381610038161003816100381610038161003816

(25)

Mathematical Problems in Engineering 9

568

568

281281

(a) 000m around the shaft wall

171

171

4343

(b) 060m around the shaft wall104

104

3737

(c) 105m around the shaft wall

85

85

5858

(d) 165m around the shaft wall

78

78

6767

(e) 255m around the shaft wall

Figure 9 von Mises stress distribution of the shaft wall in well UGS1

10 Mathematical Problems in Engineering

291

291

125125

(a) 00m around the shaft wall

89

89

1515

(b) 060m around the shaft wall

57

57

2323

(c) 105m around the shaft wall

47

47

3434

(d) 165m around the shaft wall

44

44

3838

(e) 255m around the shaft wall

Figure 10 The maximum shearing stress distribution around the shaft wall in UGS1

Mathematical Problems in Engineering 11

Table 5 Hole rock tensile fracture analysis results in UGS

Well Maximum principalstressMPa

VerticalstressMPa

Minimum principalstressMPa

PorepressureMPa

TensilestressMPa

Tensile strength[120590119905]MPa

Tensile fracturefactor 119868

120590

UGS6 minus2169 minus2597 minus2793 1200 095 497 0191UGS7 minus2132 minus2610 minus2602 1200 093 484 0192UGS8 minus2121 minus2503 minus2725 1200 099 492 0201UGS9 minus2066 minus2310 minus2783 1200 092 494 0186UGS10 minus1906 minus2515 minus2215 1200 098 495 0198

Table 6 Hole rock shear fracture analysis results in UGS

Well Maximum principalstressMPa

VerticalstressMPa

Minimum principalstressMPa

CohesivestressMPa

Maximum shearingstressMPa

Shear strength[120591119899]MPa

Shear fracturefactor 119868

119905

UGS6 minus2169 minus2597 minus2793 466 249 354 0703UGS7 minus2132 minus2610 minus2602 435 222 368 0603UGS8 minus2121 minus2503 minus2725 405 227 372 0510UGS9 minus2066 minus2310 minus2783 462 213 347 0614UGS10 minus1906 minus2515 minus2215 450 269 357 0754

The newfracture

The newfracture

x

y

120595 = minus2120573

120595 = minus2120573

120573

12059011205901

1205902

1205902

Figure 11 Fracture expansion direction of the oval fracture

where 119868120590and 119868119905are tensile fracture factor and shear fracture

factor 120590119905and 120591

119899are tensile stress and shearing stress in

the fracture plane [120590119905] and [120591

119899] are tensile strength and

shearing resistance The shaft rock will not be cracked whenthe fracture factor is below 1

72 Rock Fracture Analysis Results in the UGS According tothe Griffith fracture criterion presented above we conductthe tensile fracture analysis on the rock of the sidewall inUGS of a Turkey salt cavern With the obtained maximumand minimum horizontal in situ stresses of the test wells

r

120590h1

120590h2

120590h2

120590h1

120590120579120591r120579

120590r

Figure 12 The calculation model of wellbore stress

the tensile stress and shearing stress can be calculated bythe classical elastic theory The calculation model is shownin Figure 12 where 120590

ℎ1 120590ℎ2are the maximum and minimum

horizontal principal in situ stresses 120590119903is the radial normal

stress 120590120579is the tangential normal stress and 120591

119903120579is the

shearing stressThe results are shown in Table 5 showing that the

maximum tensile stress is smaller than the tensile strengthTherefore on the rock of the shaft wall no fracture will bedeveloped

Moreover according to the Coulomb-Navier failure cri-terion the shear fracture analysis is conducted on the rock ofthe shaft wall in UGS of a Turkey salt cavern Table 6 showsthe analysis results The shear fracture factors 119868

119905 in the hole

are all under 1 Given that themaximum shearing stress is lessthan the shearing strength the rock of the shaft wall will notdevelop a shear fracture

12 Mathematical Problems in Engineering

8 Conclusion

In this paper we improved the nonuniqueness of the in situstress field inversion solution by a priori constraints Fromthe perspective of basic equations of elastic mechanics anin situ stress numerical inversion model is built and basicfactors of in situ stress field are considered in applying thefinite element inversion method on the boundary loads ofthe in situ stress field An appropriate geometrical modeland suitable boundary conditions ensured an accurate in situstress field is derived The method is successfully applied inUGS of a Turkey salt cavern The optimisation results onprincipal stress are close to those from measurement By theoptimisation the overall in situ stress field of the researcharea is obtained and the stress distribution of the shaft wallis presented according to the calculation results The safetyof the shaft wall is confirmed by applying the correspondingfracture criterion

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to acknowledge the financial support ofthe National Natural Science Foundation of China (Grantsnos 51374228 51274231 and U1262208) and the CNPC KeyLaboratory Fund (Contract no 2014A-4214)

References

[1] B S Aadnoy ldquoInversion technique to determine the in-situstress field from fracturing datardquo Journal of Petroleum Scienceand Engineering vol 4 no 2 pp 127ndash141 1990

[2] M Bloch C A M Siqueira F H Ferreira and J C JConceicao ldquoTechniques for determining in-situ stress directionand magnitudesrdquo in Proceedings of the Latin American andCaribbean Petroleum Engineering Conference SPE 39075-MSRio de Janeiro Brazil August-September 1997

[3] J Huang D V Griffiths and S-W Wong ldquoIn situ stress deter-mination from inversion of hydraulic fracturing datardquo Interna-tional Journal of Rock Mechanics and Mining Sciences vol 48no 3 pp 476ndash481 2011

[4] J Djurhuus and B S Aadnoslashy ldquoIn situ stress state from inversionof fracturing data from oil wells and borehole image logsrdquoJournal of Petroleum Science and Engineering vol 38 no 3-4pp 121ndash130 2003

[5] B S Aadnoy ldquoIn-situ stress directions from borehole fracturetracesrdquo Journal of Petroleum Science and Engineering vol 4 no2 pp 143ndash153 1990

[6] C A Barton D Moos P Peska and M D Zoback ldquoUtilizingwellbore image data to determine the complete stress tensorapplication to permeability anisotropy and wellbore stabilityrdquoThe Log Analyst vol 38 no 6 pp 21ndash33 1997

[7] H Zhao F Ma J Xu and J Guo ldquoIn situ stress field inversionand its application in mining-induced rock mass movementrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 53 pp 120ndash128 2012

[8] Z D Chen Q A Meng and T F Wan ldquoNumerical simulationof tectonic stress field in Gulong depression in Songliao Basinusing elasitic-plastic increment methodrdquo Earth Science Fron-tiers vol 9 no 2 pp 483ndash491 2002 (Chinese)

[9] D K Singha and R Chatterjee ldquoGeomechanical modelingusing finite element method for prediction of in-situ stress inKrishna-Godavari basin Indiardquo International Journal of RockMechanics and Mining Sciences vol 73 no 5 pp 15ndash27 2015

[10] G Maier R Ardito and R Fedele ldquoInverse analysis problemsin structural engineering of concrete damsrdquo ComputationalMechanics vol 9 pp 5ndash10 2004

[11] G Maier M Bocciarelli G Bolzon and R Fedele ldquoInverseanalyses in fracture mechanicsrdquo International Journal of Frac-ture vol 138 no 1ndash4 pp 47ndash73 2006

[12] F Yin and D L Gao ldquoMechanical analysis and design of casingin directional well under in-situ stressesrdquo Journal of Natural GasScience and Engineering vol 20 pp 285ndash291 2014

[13] S Sakurai and S Akutagawa ldquoReport some aspects of backanalysis in geotechnical engineeringrdquo in Proceedings of theISRM International Symposium (EUROCK rsquo93) pp 1133ndash1140Lisboa Portugal June 1993

[14] L Z Xue Engineering Optimum Programming TechnologiesTianjing University Press Tianjing China 1998 (Chinese)

[15] F Li J-A Wang and J C Brigham ldquoInverse calculation of insitu stress in rock mass using the Surrogate-Model AcceleratedRandom Search algorithmrdquo Computers and Geotechnics vol 61pp 24ndash32 2014

[16] X Z Wang W Q Liu Y G Sun and Y J Ma ldquoField testingof fracturing techanolgy in coal formed gas wellrdquoOil Drilling ampProduction Technology vol 23 no 2 pp 58ndash61 2001 (Chinese)

[17] J Teng Y B Yao D M Liu and Y D Cai ldquoEvaluation ofcoal texture distributions in the southern Qinshui basin NorthChina investigation by amultiple geophysical loggingmethodrdquoInternational Journal of Coal Geology vol 140 pp 9ndash22 2015

[18] Z Meng J Zhang and R Wang ldquoIn-situ stress pore pressureand stress-dependent permeability in the Southern QinshuiBasinrdquo International Journal of Rock Mechanics and MiningSciences vol 48 no 1 pp 122ndash131 2011

[19] P Lancaster and K Salkauskas ldquoSurfaces generated by movingleast squaresmethodsrdquoMathematics of Computation vol 37 no155 pp 141ndash158 1981

[20] AHMedley ldquoStorage of natural gas in salt cavernsrdquo in Proceed-ings of the SPE Northern Plains Section Regional Meeting SPE-5427-MS Conference Paper Society of Petroleum EngineersOmaha Neb USA May 1975

[21] G Han M S Bruno K Lao J Young and L Dorfmann ldquoGasstorage and operations in single-bedded salt caverns stabilityanalysesrdquo SPEProduction andOperations vol 22 no 3 pp 368ndash376 2007

[22] P Berest and B Brouard ldquoLong-term behavior of salt cavernsrdquoin Proceedings of the 48th US Rock MechanicsGeomechanicsSymposium vol 14 Minneapolis Minn USA June 2014ARMA-2014-7032

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

1150 1100 1150 1150 1200 1200 1250 125012501100

1250

Tuz formation

Cihanbeyli to Kochisar formation

1100

1800

1600

1400

1200

1000

800

600

400

200

0

Dep

th b

elow

surfa

ce (m

)

UG

S12

UG

S11

UG

S10

UG

S1

UG

S2

UG

S9

UG

S8

UG

S7

UG

S6

UG

S5

UG

S4

UG

S3

Figure 3 Section map of UGS in Turkey salt cavern

UGS11

UGS10

UGS1

UGS2UGS9

UGS8

UGS7

UGS6

UGS5 UGS4UGS3

Figure 4 Three-dimensional overview of UGS in Turkey salt cavern

the in situ stress field by carefully selecting the primaryfunction and the weight function [19] The UGS distributionis shown in Figures 3ndash5 Figure 3 shows the well depths ofmultiple UGS wells The minimum depth is 1110m whilethe maximum depth is 1500m Figure 4 shows the three-dimensional distribution of these UGS wells Figure 5 showsthe planar structure of the salt cavern Twelve reservoirs asmany as UGS wells construct the salt cavern All sites are inthe southeast section which is indicated by the blue spotsThe red line in Figure 5 indicates the fault zone of the researcharea

51 Building UGS in a Salt Cavern Salt owns several proper-ties that make it ideal for gas storage [19] It has a moderatelyhigh strength and flows plastically to close fracture that couldotherwise become a leak The porosity and permeability ofsalt to liquid and gaseous hydrocarbons are close to zerothereby preventing stored gas escape [20 21] Salt caverns canprovide high deliverability and gas can be withdrawn quicklybecause no pressure from flow is lost through the pores Asalt cavern in south Turkey is used to construct undergroundnatural gas storage to compensate season-related fluctuations

of gas consumption Moreover by a salt cavern gas can beprovided to customers under optimum economic conditionsbetween buying and selling To construct UGS by a saltcavern a well should be drilled first and then water shouldbe injected to dissolve the salt Brine is pumped out of the saltcavern and a cavity is formed which can be treated to storenatural gas

52 Calculation Model and Boundary Conditions Based onthe prerequisite of the numerical simulation a geologicmodel the target layer isolation body of a rock block is takenas the simulation object Two rules should be followed todetermine the computational area (1) extend the geometricrange to eliminate or weaken the effects from the bound-ary conditions on the research area and (2) the geometryconstraint conditions on the boundary are easy to get Ingeneral the layered distribution of underground sedimentaryrock could be treated as a plane problem of elastic mechanicsin the comprehensive mathematical model Figure 5 is themechanical model of the research area which is a rectangulararea with a length of 8 km and width of 32 km Three majorfault zones are taken into consideration The lengths and

6 Mathematical Problems in Engineering

UGS11UGS12

UGS1UGS10

UGS2 UGS9UGS8 UGS7

UGS6 UGS5UGS4 UGS3

0 1000 2000(m)

Figure 5 Planar structure of the research area and the distributionof the UGS

widths of fault zones are 8 km lowast 1 km 12 km lowast 08 km and35 km lowast 08 km respectively

The following eight factors are taken as the basic factorsof tectonic force compressive or tensile forces and shearforces in the east west south and north The boundaryload is adjusted depending on the size of research area Asthe stress field is continuously changed the external stressin the circle can be divided into 12 parts which is largeenough to guarantee the accuracy of calculation By changingthe value of the 12 boundary forces we could simulate thecompressive or tensile forces in the direction of eastndashwest andsouthndashnorth as well as shear deformation in the horizontalplane to get the in situ stress field distribution of the partialresearch area At the same time boundary force constraintis applied to make the value of the resultant force and theresultant moment zero (Figure 6 shows the method to applythe boundary force)The edge points of the lower boundary ofthemodel are fixed to ensure no rigid translation and rotationin the research area

Generally a geologic body is taken as a rock with auniform block to confirm the rock physical parameters fromdifferent tectonic positions The basic assumption is thateach fault zone is homogeneous The rock strength Youngrsquosmodulus and other rock mechanical parameters should bedecreased according to the rock properties in specific depth(Table 1 shows information about mechanical parameterschosen by the model) The cored sample is taken from the

1ElementsMAT Num

Figure 6 Mechanical model of the calculated area

Table 1 Rock mechanics parameters

Number Media type Elasticity modulusMPa Poissonrsquos ratio ]1 Reservoir 31200 02252 Fault zone 1 22500 02703 Fault zone 2 23500 02604 Fault zone 3 24000 02905 Other layers 35200 0220

spot field and by indoor experiment elasticity modulus andPoissonrsquos ratio are obtained The grids of the model adoptedmainly are four-node isoperimetric rectangular elements andfor part of the area triangle elements are used In additionthe element length of the fault zone and its surrounding areais shorter and the grids are relatively finer than those of otherparts

53 Inversion of the In Situ Stress Field of the Salt Cavern

(1) Initial boundary conditions the initial values of thedistributed normal stress 119875

1sim 11987512in the east south

west and north could be set as an arbitrary value with10MPa here (Figure 6 shows how the boundary forceacts)

(2) Constraint object based on the reference about insitu stress measurement we chose 10 values of in situstress and five directions from eight survey points asknown constraint parameters to inverse 12 unknownboundary forces (Figure 5 shows the distribution ofconstraint points and measured data are shown inTables 2 and 3)

(3) Searching range the range of the boundary nor-mal stress constraint value is 1MPasim30MPa (lowerboundsimupper bound) and in the same way therange of the boundary shear stress constraint valueminus10MPasim10MPa The precision of the iteration ter-mination is 10minus5MPa2The boundary force constraintshould be taken into consideration to make sure thatthe resultant moment of the boundary force is zero

Mathematical Problems in Engineering 7

Table 2 Comparison between the inversion results and the measured values on principal stress magnitude

Measured points Stress components Inversion valueMPa Measured valueMPa Relative error

UGS6 120590119867

minus2169 minus2323 66120590ℎ

minus2793 minus2777 058

UGS7 120590119867

minus2132 minus1996 681120590ℎ

minus2602 minus261 031

UGS8 120590119867

minus2121 minus2026 469120590ℎ

minus2725 minus2852 445

UGS9 120590119867

minus2066 minus2226 719120590ℎ

minus2783 minus2664 447

UGS10 120590119867

minus1906 minus1856 269120590ℎ

minus2515 minus2385 545

Table 3 Comparison between the inverted results and themeasuredvalues on the direction of maximum principal stress

Measuredpoints Inversion value(∘) Measured value(∘) Error(∘)

UGS6 NE57327 NE55 2327UGS7 NE51798 NE535 minus1702

UGS8 NE52334 NE549 minus257

UGS9 NE4132 NE451 minus378

UGS10 NE51713 NE536 minus1887

The relevant constrained inversion model could bederived from (9)

(4) Inversion result based on the conditions above thecalculation results on the survey points t from inver-sion are shown in Tables 2 and 3 where 120590

119867and

120590ℎare the maximum and minimum principal stress

magnitude respectively The measurement resultsfrom well logging are also presented and comparedwith the inversion results Table 4 shows the results ofthe boundary load inversion

Tables 2 and 3 show that the maximum relative errorsof stress components 120590

119867and 120590

ℎare controlled within 8

The errors of the direction of maximum principal stress arecontrolled within 4∘ All the stress values obtained from theinversion reached or are close to the actual data and theprecision of the inversion results could meet engineeringrequirements

We applied the optimised boundary force from the inver-sion to perform forward calculation This approach couldlead to the distribution of the stress field value and directionof the Turkey salt cavern (shown in Figures 7 and 8)

6 Stress Distribution Character of the Wellnear the Hole of the UGS

On the basis of the analysis above the two principal stresses120590119867and 120590ℎof the 12 wells are determined Given that the stress

concentration occurs by the hole which will influence thestability of the shaft wall the von Mises stress distributioncharacter near the shaft wall (0ndash255m) should be checked

minus29

minus13minus14

minus15

minus16

minus17

minus18

minus19

minus20

minus21

minus22

minus23

minus24

minus25

minus26

minus27

minus28

minus29

minus30

minus31

minus32

minus33

minus34

minus24

minus19

minus19

minus19

minus19

minus19minus19

2000

4000

6000

8000

10000

12000

14000

4000 6000 8000 100002000

Figure 7 Distribution of the maximum principal stress and orien-tation

up as shown in Figure 9 Here well UGS1 is taken as anexample with well diameter of UGS1 279 cm bottom holepressure 12MPa overburden pressure 231MPa and welldepth 1100m

As shown in Figure 9 given that stress nonhomogeneouswith the in situ stress exists the von Mises stress distributionof the shaft wall is in an oval shape The von Mises stresshas a higher value in the shaft wall which is within 281ndash568MPa The von Mises stress decreases rapidly away fromthe shaft wall and the shape changes to a butterfly-like oneThe maximum von Mises stress decreases to 171MPa when06m away from the shaftwall It is worthmentioning that theresult here contains the influence from stress concentrationaround the shaft wall As shown in Figures 9(d) and 9(e) thevonMises stress owns the samedistribution pattern and tendsto be steady far from the shaft wall

8 Mathematical Problems in Engineering

Table 4 Optimum results of the boundary load unit MPa

1198751

1198752

1198753

1198754

1198755

1198756

1198757

1198758

1198759

11987510

11987511

11987512

2639 1999 75277 2113 2275 9216 5328 12053 1484 1021 5257 2977

2000

4000

6000

8000

10000

12000

14000

4000 6000 8000 100002000

minus24

minus38

minus43

minus43

minus43

minus38

minus43

minus43

minus33

minus38

minus28

minus33

minus38minus43

minus25

minus26

minus27

minus28

minus29

minus30

minus31

minus32

minus33

minus34

minus35

minus36

minus37

minus38

minus39

minus40

minus41

minus42

minus43

minus44

minus45

minus46

minus47

minus48

Figure 8 Distribution of the minimum principal stress and orien-tation

As shown in Figure 10 just like the von Mises stress theshearing stress distribution is in an oval shape near the shaftwall and changes to a butterfly-like shape from the shaft wallThe shearing stress on the shaft wall is maximum (291MPa)due to the stress concentration and it decreases rapidly awayfrom the shaft wall with 89MPa when 06m away Shearingstress tends to be stable when it is 16m away from the shaftwall

From the stress analysis above we can draw the conclu-sion as the following

The stress concentration phenomenon does exist aroundthe hole The von Mises stress and shearing stress aremaximum on the shaft wall The stress distribution is in anoval shape around the shaft wall while it is in a butterfly-likeshape near the hole Both stresses decrease rapidly when theyare 1m away from the shaft wall and tend to be stable when13m away from the shaft wall

7 The Hole Stability Analysis of UGS

71 The Rock Fracture Analysis around the Hole of UGSBased on the Griffith Principle Griffith defined the inherentdefects or micro-fracture in the material as the Griffithfracture Based on theGriffith criterion [22] the rock fracturecondition in the planar stress state is shown below

When (1205901+ 31205902) gt 0 the fracture criterion is

(1205902minus 1205901)2= 8 (120590

2+ 1205901) 120590119905 (20)

Equation (20) can also be like 120590119905= (1205902minus 1205901)28(1205902+ 1205901)

cos 2120573 = (1205901minus 1205902)2(1205902+ 1205901)

When (1205901+ 31205902) le 0 the fracture criterion is

1205902= minus120590119905

sin 2120573 = 0

(21)

where 120590119905is the tensile stress from the rock unidirectional

tensile test 120573 is the angle between the long-axis of theoval fracture and the principal compressive stress axis Thefracture angle is 120579 = 120573

If a new fracture occurs it will be in the direction of ovalfractureThe angle between the new fracture and the originalone is 120595 (Figure 11) The following equation can be obtainedby the fracture criterion

120595 = minus2120573 (22)

Equation (22) shows that the angle between the newfracture and the original one is 2120573 in the condition of (120590

1+

31205902) gt 0 The negative sign indicates the direction of the

new fracture is clockwise rotation by 120573 from the principalcompression stress axis (Figure 11)

For the three-dimensional problem Griffith fracturecriterion can be depicted as follows

When (1205901+ 31205903) gt 0 the fracture criterion is

(1205901minus 1205902)2+ (1205902minus 1205903)2+ (1205903minus 1205901)2

= 24120590119905(1205901+ 1205902+ 1205903)

cos 2120579 =1205901minus 1205903

2 (1205901+ 1205903)

(23)

When (1205901+ 31205903) le 0 the fracture criterion is

120590119905= minus1205903 (24)

Based on the fracture criterion above it can be judgedonly whether the tensile fracture or the shear fracture occurswhile the developing degrees of the fracture cannot bedetermined In fact with the asymmetry of the tectonic stressdistribution the fracture developing degrees of tectonic partdiffer The concept of fracture factor is introduced to judgethe fracture developing degrees of the rock quantitativelyThefracture factor is given by

119868120590=

100381610038161003816100381610038161003816100381610038161003816

120590119905

[120590119905]

100381610038161003816100381610038161003816100381610038161003816

or 119868119905=

100381610038161003816100381610038161003816100381610038161003816

120591119899

[120591119899]

100381610038161003816100381610038161003816100381610038161003816

(25)

Mathematical Problems in Engineering 9

568

568

281281

(a) 000m around the shaft wall

171

171

4343

(b) 060m around the shaft wall104

104

3737

(c) 105m around the shaft wall

85

85

5858

(d) 165m around the shaft wall

78

78

6767

(e) 255m around the shaft wall

Figure 9 von Mises stress distribution of the shaft wall in well UGS1

10 Mathematical Problems in Engineering

291

291

125125

(a) 00m around the shaft wall

89

89

1515

(b) 060m around the shaft wall

57

57

2323

(c) 105m around the shaft wall

47

47

3434

(d) 165m around the shaft wall

44

44

3838

(e) 255m around the shaft wall

Figure 10 The maximum shearing stress distribution around the shaft wall in UGS1

Mathematical Problems in Engineering 11

Table 5 Hole rock tensile fracture analysis results in UGS

Well Maximum principalstressMPa

VerticalstressMPa

Minimum principalstressMPa

PorepressureMPa

TensilestressMPa

Tensile strength[120590119905]MPa

Tensile fracturefactor 119868

120590

UGS6 minus2169 minus2597 minus2793 1200 095 497 0191UGS7 minus2132 minus2610 minus2602 1200 093 484 0192UGS8 minus2121 minus2503 minus2725 1200 099 492 0201UGS9 minus2066 minus2310 minus2783 1200 092 494 0186UGS10 minus1906 minus2515 minus2215 1200 098 495 0198

Table 6 Hole rock shear fracture analysis results in UGS

Well Maximum principalstressMPa

VerticalstressMPa

Minimum principalstressMPa

CohesivestressMPa

Maximum shearingstressMPa

Shear strength[120591119899]MPa

Shear fracturefactor 119868

119905

UGS6 minus2169 minus2597 minus2793 466 249 354 0703UGS7 minus2132 minus2610 minus2602 435 222 368 0603UGS8 minus2121 minus2503 minus2725 405 227 372 0510UGS9 minus2066 minus2310 minus2783 462 213 347 0614UGS10 minus1906 minus2515 minus2215 450 269 357 0754

The newfracture

The newfracture

x

y

120595 = minus2120573

120595 = minus2120573

120573

12059011205901

1205902

1205902

Figure 11 Fracture expansion direction of the oval fracture

where 119868120590and 119868119905are tensile fracture factor and shear fracture

factor 120590119905and 120591

119899are tensile stress and shearing stress in

the fracture plane [120590119905] and [120591

119899] are tensile strength and

shearing resistance The shaft rock will not be cracked whenthe fracture factor is below 1

72 Rock Fracture Analysis Results in the UGS According tothe Griffith fracture criterion presented above we conductthe tensile fracture analysis on the rock of the sidewall inUGS of a Turkey salt cavern With the obtained maximumand minimum horizontal in situ stresses of the test wells

r

120590h1

120590h2

120590h2

120590h1

120590120579120591r120579

120590r

Figure 12 The calculation model of wellbore stress

the tensile stress and shearing stress can be calculated bythe classical elastic theory The calculation model is shownin Figure 12 where 120590

ℎ1 120590ℎ2are the maximum and minimum

horizontal principal in situ stresses 120590119903is the radial normal

stress 120590120579is the tangential normal stress and 120591

119903120579is the

shearing stressThe results are shown in Table 5 showing that the

maximum tensile stress is smaller than the tensile strengthTherefore on the rock of the shaft wall no fracture will bedeveloped

Moreover according to the Coulomb-Navier failure cri-terion the shear fracture analysis is conducted on the rock ofthe shaft wall in UGS of a Turkey salt cavern Table 6 showsthe analysis results The shear fracture factors 119868

119905 in the hole

are all under 1 Given that themaximum shearing stress is lessthan the shearing strength the rock of the shaft wall will notdevelop a shear fracture

12 Mathematical Problems in Engineering

8 Conclusion

In this paper we improved the nonuniqueness of the in situstress field inversion solution by a priori constraints Fromthe perspective of basic equations of elastic mechanics anin situ stress numerical inversion model is built and basicfactors of in situ stress field are considered in applying thefinite element inversion method on the boundary loads ofthe in situ stress field An appropriate geometrical modeland suitable boundary conditions ensured an accurate in situstress field is derived The method is successfully applied inUGS of a Turkey salt cavern The optimisation results onprincipal stress are close to those from measurement By theoptimisation the overall in situ stress field of the researcharea is obtained and the stress distribution of the shaft wallis presented according to the calculation results The safetyof the shaft wall is confirmed by applying the correspondingfracture criterion

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to acknowledge the financial support ofthe National Natural Science Foundation of China (Grantsnos 51374228 51274231 and U1262208) and the CNPC KeyLaboratory Fund (Contract no 2014A-4214)

References

[1] B S Aadnoy ldquoInversion technique to determine the in-situstress field from fracturing datardquo Journal of Petroleum Scienceand Engineering vol 4 no 2 pp 127ndash141 1990

[2] M Bloch C A M Siqueira F H Ferreira and J C JConceicao ldquoTechniques for determining in-situ stress directionand magnitudesrdquo in Proceedings of the Latin American andCaribbean Petroleum Engineering Conference SPE 39075-MSRio de Janeiro Brazil August-September 1997

[3] J Huang D V Griffiths and S-W Wong ldquoIn situ stress deter-mination from inversion of hydraulic fracturing datardquo Interna-tional Journal of Rock Mechanics and Mining Sciences vol 48no 3 pp 476ndash481 2011

[4] J Djurhuus and B S Aadnoslashy ldquoIn situ stress state from inversionof fracturing data from oil wells and borehole image logsrdquoJournal of Petroleum Science and Engineering vol 38 no 3-4pp 121ndash130 2003

[5] B S Aadnoy ldquoIn-situ stress directions from borehole fracturetracesrdquo Journal of Petroleum Science and Engineering vol 4 no2 pp 143ndash153 1990

[6] C A Barton D Moos P Peska and M D Zoback ldquoUtilizingwellbore image data to determine the complete stress tensorapplication to permeability anisotropy and wellbore stabilityrdquoThe Log Analyst vol 38 no 6 pp 21ndash33 1997

[7] H Zhao F Ma J Xu and J Guo ldquoIn situ stress field inversionand its application in mining-induced rock mass movementrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 53 pp 120ndash128 2012

[8] Z D Chen Q A Meng and T F Wan ldquoNumerical simulationof tectonic stress field in Gulong depression in Songliao Basinusing elasitic-plastic increment methodrdquo Earth Science Fron-tiers vol 9 no 2 pp 483ndash491 2002 (Chinese)

[9] D K Singha and R Chatterjee ldquoGeomechanical modelingusing finite element method for prediction of in-situ stress inKrishna-Godavari basin Indiardquo International Journal of RockMechanics and Mining Sciences vol 73 no 5 pp 15ndash27 2015

[10] G Maier R Ardito and R Fedele ldquoInverse analysis problemsin structural engineering of concrete damsrdquo ComputationalMechanics vol 9 pp 5ndash10 2004

[11] G Maier M Bocciarelli G Bolzon and R Fedele ldquoInverseanalyses in fracture mechanicsrdquo International Journal of Frac-ture vol 138 no 1ndash4 pp 47ndash73 2006

[12] F Yin and D L Gao ldquoMechanical analysis and design of casingin directional well under in-situ stressesrdquo Journal of Natural GasScience and Engineering vol 20 pp 285ndash291 2014

[13] S Sakurai and S Akutagawa ldquoReport some aspects of backanalysis in geotechnical engineeringrdquo in Proceedings of theISRM International Symposium (EUROCK rsquo93) pp 1133ndash1140Lisboa Portugal June 1993

[14] L Z Xue Engineering Optimum Programming TechnologiesTianjing University Press Tianjing China 1998 (Chinese)

[15] F Li J-A Wang and J C Brigham ldquoInverse calculation of insitu stress in rock mass using the Surrogate-Model AcceleratedRandom Search algorithmrdquo Computers and Geotechnics vol 61pp 24ndash32 2014

[16] X Z Wang W Q Liu Y G Sun and Y J Ma ldquoField testingof fracturing techanolgy in coal formed gas wellrdquoOil Drilling ampProduction Technology vol 23 no 2 pp 58ndash61 2001 (Chinese)

[17] J Teng Y B Yao D M Liu and Y D Cai ldquoEvaluation ofcoal texture distributions in the southern Qinshui basin NorthChina investigation by amultiple geophysical loggingmethodrdquoInternational Journal of Coal Geology vol 140 pp 9ndash22 2015

[18] Z Meng J Zhang and R Wang ldquoIn-situ stress pore pressureand stress-dependent permeability in the Southern QinshuiBasinrdquo International Journal of Rock Mechanics and MiningSciences vol 48 no 1 pp 122ndash131 2011

[19] P Lancaster and K Salkauskas ldquoSurfaces generated by movingleast squaresmethodsrdquoMathematics of Computation vol 37 no155 pp 141ndash158 1981

[20] AHMedley ldquoStorage of natural gas in salt cavernsrdquo in Proceed-ings of the SPE Northern Plains Section Regional Meeting SPE-5427-MS Conference Paper Society of Petroleum EngineersOmaha Neb USA May 1975

[21] G Han M S Bruno K Lao J Young and L Dorfmann ldquoGasstorage and operations in single-bedded salt caverns stabilityanalysesrdquo SPEProduction andOperations vol 22 no 3 pp 368ndash376 2007

[22] P Berest and B Brouard ldquoLong-term behavior of salt cavernsrdquoin Proceedings of the 48th US Rock MechanicsGeomechanicsSymposium vol 14 Minneapolis Minn USA June 2014ARMA-2014-7032

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

UGS11UGS12

UGS1UGS10

UGS2 UGS9UGS8 UGS7

UGS6 UGS5UGS4 UGS3

0 1000 2000(m)

Figure 5 Planar structure of the research area and the distributionof the UGS

widths of fault zones are 8 km lowast 1 km 12 km lowast 08 km and35 km lowast 08 km respectively

The following eight factors are taken as the basic factorsof tectonic force compressive or tensile forces and shearforces in the east west south and north The boundaryload is adjusted depending on the size of research area Asthe stress field is continuously changed the external stressin the circle can be divided into 12 parts which is largeenough to guarantee the accuracy of calculation By changingthe value of the 12 boundary forces we could simulate thecompressive or tensile forces in the direction of eastndashwest andsouthndashnorth as well as shear deformation in the horizontalplane to get the in situ stress field distribution of the partialresearch area At the same time boundary force constraintis applied to make the value of the resultant force and theresultant moment zero (Figure 6 shows the method to applythe boundary force)The edge points of the lower boundary ofthemodel are fixed to ensure no rigid translation and rotationin the research area

Generally a geologic body is taken as a rock with auniform block to confirm the rock physical parameters fromdifferent tectonic positions The basic assumption is thateach fault zone is homogeneous The rock strength Youngrsquosmodulus and other rock mechanical parameters should bedecreased according to the rock properties in specific depth(Table 1 shows information about mechanical parameterschosen by the model) The cored sample is taken from the

1ElementsMAT Num

Figure 6 Mechanical model of the calculated area

Table 1 Rock mechanics parameters

Number Media type Elasticity modulusMPa Poissonrsquos ratio ]1 Reservoir 31200 02252 Fault zone 1 22500 02703 Fault zone 2 23500 02604 Fault zone 3 24000 02905 Other layers 35200 0220

spot field and by indoor experiment elasticity modulus andPoissonrsquos ratio are obtained The grids of the model adoptedmainly are four-node isoperimetric rectangular elements andfor part of the area triangle elements are used In additionthe element length of the fault zone and its surrounding areais shorter and the grids are relatively finer than those of otherparts

53 Inversion of the In Situ Stress Field of the Salt Cavern

(1) Initial boundary conditions the initial values of thedistributed normal stress 119875

1sim 11987512in the east south

west and north could be set as an arbitrary value with10MPa here (Figure 6 shows how the boundary forceacts)

(2) Constraint object based on the reference about insitu stress measurement we chose 10 values of in situstress and five directions from eight survey points asknown constraint parameters to inverse 12 unknownboundary forces (Figure 5 shows the distribution ofconstraint points and measured data are shown inTables 2 and 3)

(3) Searching range the range of the boundary nor-mal stress constraint value is 1MPasim30MPa (lowerboundsimupper bound) and in the same way therange of the boundary shear stress constraint valueminus10MPasim10MPa The precision of the iteration ter-mination is 10minus5MPa2The boundary force constraintshould be taken into consideration to make sure thatthe resultant moment of the boundary force is zero

Mathematical Problems in Engineering 7

Table 2 Comparison between the inversion results and the measured values on principal stress magnitude

Measured points Stress components Inversion valueMPa Measured valueMPa Relative error

UGS6 120590119867

minus2169 minus2323 66120590ℎ

minus2793 minus2777 058

UGS7 120590119867

minus2132 minus1996 681120590ℎ

minus2602 minus261 031

UGS8 120590119867

minus2121 minus2026 469120590ℎ

minus2725 minus2852 445

UGS9 120590119867

minus2066 minus2226 719120590ℎ

minus2783 minus2664 447

UGS10 120590119867

minus1906 minus1856 269120590ℎ

minus2515 minus2385 545

Table 3 Comparison between the inverted results and themeasuredvalues on the direction of maximum principal stress

Measuredpoints Inversion value(∘) Measured value(∘) Error(∘)

UGS6 NE57327 NE55 2327UGS7 NE51798 NE535 minus1702

UGS8 NE52334 NE549 minus257

UGS9 NE4132 NE451 minus378

UGS10 NE51713 NE536 minus1887

The relevant constrained inversion model could bederived from (9)

(4) Inversion result based on the conditions above thecalculation results on the survey points t from inver-sion are shown in Tables 2 and 3 where 120590

119867and

120590ℎare the maximum and minimum principal stress

magnitude respectively The measurement resultsfrom well logging are also presented and comparedwith the inversion results Table 4 shows the results ofthe boundary load inversion

Tables 2 and 3 show that the maximum relative errorsof stress components 120590

119867and 120590

ℎare controlled within 8

The errors of the direction of maximum principal stress arecontrolled within 4∘ All the stress values obtained from theinversion reached or are close to the actual data and theprecision of the inversion results could meet engineeringrequirements

We applied the optimised boundary force from the inver-sion to perform forward calculation This approach couldlead to the distribution of the stress field value and directionof the Turkey salt cavern (shown in Figures 7 and 8)

6 Stress Distribution Character of the Wellnear the Hole of the UGS

On the basis of the analysis above the two principal stresses120590119867and 120590ℎof the 12 wells are determined Given that the stress

concentration occurs by the hole which will influence thestability of the shaft wall the von Mises stress distributioncharacter near the shaft wall (0ndash255m) should be checked

minus29

minus13minus14

minus15

minus16

minus17

minus18

minus19

minus20

minus21

minus22

minus23

minus24

minus25

minus26

minus27

minus28

minus29

minus30

minus31

minus32

minus33

minus34

minus24

minus19

minus19

minus19

minus19

minus19minus19

2000

4000

6000

8000

10000

12000

14000

4000 6000 8000 100002000

Figure 7 Distribution of the maximum principal stress and orien-tation

up as shown in Figure 9 Here well UGS1 is taken as anexample with well diameter of UGS1 279 cm bottom holepressure 12MPa overburden pressure 231MPa and welldepth 1100m

As shown in Figure 9 given that stress nonhomogeneouswith the in situ stress exists the von Mises stress distributionof the shaft wall is in an oval shape The von Mises stresshas a higher value in the shaft wall which is within 281ndash568MPa The von Mises stress decreases rapidly away fromthe shaft wall and the shape changes to a butterfly-like oneThe maximum von Mises stress decreases to 171MPa when06m away from the shaftwall It is worthmentioning that theresult here contains the influence from stress concentrationaround the shaft wall As shown in Figures 9(d) and 9(e) thevonMises stress owns the samedistribution pattern and tendsto be steady far from the shaft wall

8 Mathematical Problems in Engineering

Table 4 Optimum results of the boundary load unit MPa

1198751

1198752

1198753

1198754

1198755

1198756

1198757

1198758

1198759

11987510

11987511

11987512

2639 1999 75277 2113 2275 9216 5328 12053 1484 1021 5257 2977

2000

4000

6000

8000

10000

12000

14000

4000 6000 8000 100002000

minus24

minus38

minus43

minus43

minus43

minus38

minus43

minus43

minus33

minus38

minus28

minus33

minus38minus43

minus25

minus26

minus27

minus28

minus29

minus30

minus31

minus32

minus33

minus34

minus35

minus36

minus37

minus38

minus39

minus40

minus41

minus42

minus43

minus44

minus45

minus46

minus47

minus48

Figure 8 Distribution of the minimum principal stress and orien-tation

As shown in Figure 10 just like the von Mises stress theshearing stress distribution is in an oval shape near the shaftwall and changes to a butterfly-like shape from the shaft wallThe shearing stress on the shaft wall is maximum (291MPa)due to the stress concentration and it decreases rapidly awayfrom the shaft wall with 89MPa when 06m away Shearingstress tends to be stable when it is 16m away from the shaftwall

From the stress analysis above we can draw the conclu-sion as the following

The stress concentration phenomenon does exist aroundthe hole The von Mises stress and shearing stress aremaximum on the shaft wall The stress distribution is in anoval shape around the shaft wall while it is in a butterfly-likeshape near the hole Both stresses decrease rapidly when theyare 1m away from the shaft wall and tend to be stable when13m away from the shaft wall

7 The Hole Stability Analysis of UGS

71 The Rock Fracture Analysis around the Hole of UGSBased on the Griffith Principle Griffith defined the inherentdefects or micro-fracture in the material as the Griffithfracture Based on theGriffith criterion [22] the rock fracturecondition in the planar stress state is shown below

When (1205901+ 31205902) gt 0 the fracture criterion is

(1205902minus 1205901)2= 8 (120590

2+ 1205901) 120590119905 (20)

Equation (20) can also be like 120590119905= (1205902minus 1205901)28(1205902+ 1205901)

cos 2120573 = (1205901minus 1205902)2(1205902+ 1205901)

When (1205901+ 31205902) le 0 the fracture criterion is

1205902= minus120590119905

sin 2120573 = 0

(21)

where 120590119905is the tensile stress from the rock unidirectional

tensile test 120573 is the angle between the long-axis of theoval fracture and the principal compressive stress axis Thefracture angle is 120579 = 120573

If a new fracture occurs it will be in the direction of ovalfractureThe angle between the new fracture and the originalone is 120595 (Figure 11) The following equation can be obtainedby the fracture criterion

120595 = minus2120573 (22)

Equation (22) shows that the angle between the newfracture and the original one is 2120573 in the condition of (120590

1+

31205902) gt 0 The negative sign indicates the direction of the

new fracture is clockwise rotation by 120573 from the principalcompression stress axis (Figure 11)

For the three-dimensional problem Griffith fracturecriterion can be depicted as follows

When (1205901+ 31205903) gt 0 the fracture criterion is

(1205901minus 1205902)2+ (1205902minus 1205903)2+ (1205903minus 1205901)2

= 24120590119905(1205901+ 1205902+ 1205903)

cos 2120579 =1205901minus 1205903

2 (1205901+ 1205903)

(23)

When (1205901+ 31205903) le 0 the fracture criterion is

120590119905= minus1205903 (24)

Based on the fracture criterion above it can be judgedonly whether the tensile fracture or the shear fracture occurswhile the developing degrees of the fracture cannot bedetermined In fact with the asymmetry of the tectonic stressdistribution the fracture developing degrees of tectonic partdiffer The concept of fracture factor is introduced to judgethe fracture developing degrees of the rock quantitativelyThefracture factor is given by

119868120590=

100381610038161003816100381610038161003816100381610038161003816

120590119905

[120590119905]

100381610038161003816100381610038161003816100381610038161003816

or 119868119905=

100381610038161003816100381610038161003816100381610038161003816

120591119899

[120591119899]

100381610038161003816100381610038161003816100381610038161003816

(25)

Mathematical Problems in Engineering 9

568

568

281281

(a) 000m around the shaft wall

171

171

4343

(b) 060m around the shaft wall104

104

3737

(c) 105m around the shaft wall

85

85

5858

(d) 165m around the shaft wall

78

78

6767

(e) 255m around the shaft wall

Figure 9 von Mises stress distribution of the shaft wall in well UGS1

10 Mathematical Problems in Engineering

291

291

125125

(a) 00m around the shaft wall

89

89

1515

(b) 060m around the shaft wall

57

57

2323

(c) 105m around the shaft wall

47

47

3434

(d) 165m around the shaft wall

44

44

3838

(e) 255m around the shaft wall

Figure 10 The maximum shearing stress distribution around the shaft wall in UGS1

Mathematical Problems in Engineering 11

Table 5 Hole rock tensile fracture analysis results in UGS

Well Maximum principalstressMPa

VerticalstressMPa

Minimum principalstressMPa

PorepressureMPa

TensilestressMPa

Tensile strength[120590119905]MPa

Tensile fracturefactor 119868

120590

UGS6 minus2169 minus2597 minus2793 1200 095 497 0191UGS7 minus2132 minus2610 minus2602 1200 093 484 0192UGS8 minus2121 minus2503 minus2725 1200 099 492 0201UGS9 minus2066 minus2310 minus2783 1200 092 494 0186UGS10 minus1906 minus2515 minus2215 1200 098 495 0198

Table 6 Hole rock shear fracture analysis results in UGS

Well Maximum principalstressMPa

VerticalstressMPa

Minimum principalstressMPa

CohesivestressMPa

Maximum shearingstressMPa

Shear strength[120591119899]MPa

Shear fracturefactor 119868

119905

UGS6 minus2169 minus2597 minus2793 466 249 354 0703UGS7 minus2132 minus2610 minus2602 435 222 368 0603UGS8 minus2121 minus2503 minus2725 405 227 372 0510UGS9 minus2066 minus2310 minus2783 462 213 347 0614UGS10 minus1906 minus2515 minus2215 450 269 357 0754

The newfracture

The newfracture

x

y

120595 = minus2120573

120595 = minus2120573

120573

12059011205901

1205902

1205902

Figure 11 Fracture expansion direction of the oval fracture

where 119868120590and 119868119905are tensile fracture factor and shear fracture

factor 120590119905and 120591

119899are tensile stress and shearing stress in

the fracture plane [120590119905] and [120591

119899] are tensile strength and

shearing resistance The shaft rock will not be cracked whenthe fracture factor is below 1

72 Rock Fracture Analysis Results in the UGS According tothe Griffith fracture criterion presented above we conductthe tensile fracture analysis on the rock of the sidewall inUGS of a Turkey salt cavern With the obtained maximumand minimum horizontal in situ stresses of the test wells

r

120590h1

120590h2

120590h2

120590h1

120590120579120591r120579

120590r

Figure 12 The calculation model of wellbore stress

the tensile stress and shearing stress can be calculated bythe classical elastic theory The calculation model is shownin Figure 12 where 120590

ℎ1 120590ℎ2are the maximum and minimum

horizontal principal in situ stresses 120590119903is the radial normal

stress 120590120579is the tangential normal stress and 120591

119903120579is the

shearing stressThe results are shown in Table 5 showing that the

maximum tensile stress is smaller than the tensile strengthTherefore on the rock of the shaft wall no fracture will bedeveloped

Moreover according to the Coulomb-Navier failure cri-terion the shear fracture analysis is conducted on the rock ofthe shaft wall in UGS of a Turkey salt cavern Table 6 showsthe analysis results The shear fracture factors 119868

119905 in the hole

are all under 1 Given that themaximum shearing stress is lessthan the shearing strength the rock of the shaft wall will notdevelop a shear fracture

12 Mathematical Problems in Engineering

8 Conclusion

In this paper we improved the nonuniqueness of the in situstress field inversion solution by a priori constraints Fromthe perspective of basic equations of elastic mechanics anin situ stress numerical inversion model is built and basicfactors of in situ stress field are considered in applying thefinite element inversion method on the boundary loads ofthe in situ stress field An appropriate geometrical modeland suitable boundary conditions ensured an accurate in situstress field is derived The method is successfully applied inUGS of a Turkey salt cavern The optimisation results onprincipal stress are close to those from measurement By theoptimisation the overall in situ stress field of the researcharea is obtained and the stress distribution of the shaft wallis presented according to the calculation results The safetyof the shaft wall is confirmed by applying the correspondingfracture criterion

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to acknowledge the financial support ofthe National Natural Science Foundation of China (Grantsnos 51374228 51274231 and U1262208) and the CNPC KeyLaboratory Fund (Contract no 2014A-4214)

References

[1] B S Aadnoy ldquoInversion technique to determine the in-situstress field from fracturing datardquo Journal of Petroleum Scienceand Engineering vol 4 no 2 pp 127ndash141 1990

[2] M Bloch C A M Siqueira F H Ferreira and J C JConceicao ldquoTechniques for determining in-situ stress directionand magnitudesrdquo in Proceedings of the Latin American andCaribbean Petroleum Engineering Conference SPE 39075-MSRio de Janeiro Brazil August-September 1997

[3] J Huang D V Griffiths and S-W Wong ldquoIn situ stress deter-mination from inversion of hydraulic fracturing datardquo Interna-tional Journal of Rock Mechanics and Mining Sciences vol 48no 3 pp 476ndash481 2011

[4] J Djurhuus and B S Aadnoslashy ldquoIn situ stress state from inversionof fracturing data from oil wells and borehole image logsrdquoJournal of Petroleum Science and Engineering vol 38 no 3-4pp 121ndash130 2003

[5] B S Aadnoy ldquoIn-situ stress directions from borehole fracturetracesrdquo Journal of Petroleum Science and Engineering vol 4 no2 pp 143ndash153 1990

[6] C A Barton D Moos P Peska and M D Zoback ldquoUtilizingwellbore image data to determine the complete stress tensorapplication to permeability anisotropy and wellbore stabilityrdquoThe Log Analyst vol 38 no 6 pp 21ndash33 1997

[7] H Zhao F Ma J Xu and J Guo ldquoIn situ stress field inversionand its application in mining-induced rock mass movementrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 53 pp 120ndash128 2012

[8] Z D Chen Q A Meng and T F Wan ldquoNumerical simulationof tectonic stress field in Gulong depression in Songliao Basinusing elasitic-plastic increment methodrdquo Earth Science Fron-tiers vol 9 no 2 pp 483ndash491 2002 (Chinese)

[9] D K Singha and R Chatterjee ldquoGeomechanical modelingusing finite element method for prediction of in-situ stress inKrishna-Godavari basin Indiardquo International Journal of RockMechanics and Mining Sciences vol 73 no 5 pp 15ndash27 2015

[10] G Maier R Ardito and R Fedele ldquoInverse analysis problemsin structural engineering of concrete damsrdquo ComputationalMechanics vol 9 pp 5ndash10 2004

[11] G Maier M Bocciarelli G Bolzon and R Fedele ldquoInverseanalyses in fracture mechanicsrdquo International Journal of Frac-ture vol 138 no 1ndash4 pp 47ndash73 2006

[12] F Yin and D L Gao ldquoMechanical analysis and design of casingin directional well under in-situ stressesrdquo Journal of Natural GasScience and Engineering vol 20 pp 285ndash291 2014

[13] S Sakurai and S Akutagawa ldquoReport some aspects of backanalysis in geotechnical engineeringrdquo in Proceedings of theISRM International Symposium (EUROCK rsquo93) pp 1133ndash1140Lisboa Portugal June 1993

[14] L Z Xue Engineering Optimum Programming TechnologiesTianjing University Press Tianjing China 1998 (Chinese)

[15] F Li J-A Wang and J C Brigham ldquoInverse calculation of insitu stress in rock mass using the Surrogate-Model AcceleratedRandom Search algorithmrdquo Computers and Geotechnics vol 61pp 24ndash32 2014

[16] X Z Wang W Q Liu Y G Sun and Y J Ma ldquoField testingof fracturing techanolgy in coal formed gas wellrdquoOil Drilling ampProduction Technology vol 23 no 2 pp 58ndash61 2001 (Chinese)

[17] J Teng Y B Yao D M Liu and Y D Cai ldquoEvaluation ofcoal texture distributions in the southern Qinshui basin NorthChina investigation by amultiple geophysical loggingmethodrdquoInternational Journal of Coal Geology vol 140 pp 9ndash22 2015

[18] Z Meng J Zhang and R Wang ldquoIn-situ stress pore pressureand stress-dependent permeability in the Southern QinshuiBasinrdquo International Journal of Rock Mechanics and MiningSciences vol 48 no 1 pp 122ndash131 2011

[19] P Lancaster and K Salkauskas ldquoSurfaces generated by movingleast squaresmethodsrdquoMathematics of Computation vol 37 no155 pp 141ndash158 1981

[20] AHMedley ldquoStorage of natural gas in salt cavernsrdquo in Proceed-ings of the SPE Northern Plains Section Regional Meeting SPE-5427-MS Conference Paper Society of Petroleum EngineersOmaha Neb USA May 1975

[21] G Han M S Bruno K Lao J Young and L Dorfmann ldquoGasstorage and operations in single-bedded salt caverns stabilityanalysesrdquo SPEProduction andOperations vol 22 no 3 pp 368ndash376 2007

[22] P Berest and B Brouard ldquoLong-term behavior of salt cavernsrdquoin Proceedings of the 48th US Rock MechanicsGeomechanicsSymposium vol 14 Minneapolis Minn USA June 2014ARMA-2014-7032

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Table 2 Comparison between the inversion results and the measured values on principal stress magnitude

Measured points Stress components Inversion valueMPa Measured valueMPa Relative error

UGS6 120590119867

minus2169 minus2323 66120590ℎ

minus2793 minus2777 058

UGS7 120590119867

minus2132 minus1996 681120590ℎ

minus2602 minus261 031

UGS8 120590119867

minus2121 minus2026 469120590ℎ

minus2725 minus2852 445

UGS9 120590119867

minus2066 minus2226 719120590ℎ

minus2783 minus2664 447

UGS10 120590119867

minus1906 minus1856 269120590ℎ

minus2515 minus2385 545

Table 3 Comparison between the inverted results and themeasuredvalues on the direction of maximum principal stress

Measuredpoints Inversion value(∘) Measured value(∘) Error(∘)

UGS6 NE57327 NE55 2327UGS7 NE51798 NE535 minus1702

UGS8 NE52334 NE549 minus257

UGS9 NE4132 NE451 minus378

UGS10 NE51713 NE536 minus1887

The relevant constrained inversion model could bederived from (9)

(4) Inversion result based on the conditions above thecalculation results on the survey points t from inver-sion are shown in Tables 2 and 3 where 120590

119867and

120590ℎare the maximum and minimum principal stress

magnitude respectively The measurement resultsfrom well logging are also presented and comparedwith the inversion results Table 4 shows the results ofthe boundary load inversion

Tables 2 and 3 show that the maximum relative errorsof stress components 120590

119867and 120590

ℎare controlled within 8

The errors of the direction of maximum principal stress arecontrolled within 4∘ All the stress values obtained from theinversion reached or are close to the actual data and theprecision of the inversion results could meet engineeringrequirements

We applied the optimised boundary force from the inver-sion to perform forward calculation This approach couldlead to the distribution of the stress field value and directionof the Turkey salt cavern (shown in Figures 7 and 8)

6 Stress Distribution Character of the Wellnear the Hole of the UGS

On the basis of the analysis above the two principal stresses120590119867and 120590ℎof the 12 wells are determined Given that the stress

concentration occurs by the hole which will influence thestability of the shaft wall the von Mises stress distributioncharacter near the shaft wall (0ndash255m) should be checked

minus29

minus13minus14

minus15

minus16

minus17

minus18

minus19

minus20

minus21

minus22

minus23

minus24

minus25

minus26

minus27

minus28

minus29

minus30

minus31

minus32

minus33

minus34

minus24

minus19

minus19

minus19

minus19

minus19minus19

2000

4000

6000

8000

10000

12000

14000

4000 6000 8000 100002000

Figure 7 Distribution of the maximum principal stress and orien-tation

up as shown in Figure 9 Here well UGS1 is taken as anexample with well diameter of UGS1 279 cm bottom holepressure 12MPa overburden pressure 231MPa and welldepth 1100m

As shown in Figure 9 given that stress nonhomogeneouswith the in situ stress exists the von Mises stress distributionof the shaft wall is in an oval shape The von Mises stresshas a higher value in the shaft wall which is within 281ndash568MPa The von Mises stress decreases rapidly away fromthe shaft wall and the shape changes to a butterfly-like oneThe maximum von Mises stress decreases to 171MPa when06m away from the shaftwall It is worthmentioning that theresult here contains the influence from stress concentrationaround the shaft wall As shown in Figures 9(d) and 9(e) thevonMises stress owns the samedistribution pattern and tendsto be steady far from the shaft wall

8 Mathematical Problems in Engineering

Table 4 Optimum results of the boundary load unit MPa

1198751

1198752

1198753

1198754

1198755

1198756

1198757

1198758

1198759

11987510

11987511

11987512

2639 1999 75277 2113 2275 9216 5328 12053 1484 1021 5257 2977

2000

4000

6000

8000

10000

12000

14000

4000 6000 8000 100002000

minus24

minus38

minus43

minus43

minus43

minus38

minus43

minus43

minus33

minus38

minus28

minus33

minus38minus43

minus25

minus26

minus27

minus28

minus29

minus30

minus31

minus32

minus33

minus34

minus35

minus36

minus37

minus38

minus39

minus40

minus41

minus42

minus43

minus44

minus45

minus46

minus47

minus48

Figure 8 Distribution of the minimum principal stress and orien-tation

As shown in Figure 10 just like the von Mises stress theshearing stress distribution is in an oval shape near the shaftwall and changes to a butterfly-like shape from the shaft wallThe shearing stress on the shaft wall is maximum (291MPa)due to the stress concentration and it decreases rapidly awayfrom the shaft wall with 89MPa when 06m away Shearingstress tends to be stable when it is 16m away from the shaftwall

From the stress analysis above we can draw the conclu-sion as the following

The stress concentration phenomenon does exist aroundthe hole The von Mises stress and shearing stress aremaximum on the shaft wall The stress distribution is in anoval shape around the shaft wall while it is in a butterfly-likeshape near the hole Both stresses decrease rapidly when theyare 1m away from the shaft wall and tend to be stable when13m away from the shaft wall

7 The Hole Stability Analysis of UGS

71 The Rock Fracture Analysis around the Hole of UGSBased on the Griffith Principle Griffith defined the inherentdefects or micro-fracture in the material as the Griffithfracture Based on theGriffith criterion [22] the rock fracturecondition in the planar stress state is shown below

When (1205901+ 31205902) gt 0 the fracture criterion is

(1205902minus 1205901)2= 8 (120590

2+ 1205901) 120590119905 (20)

Equation (20) can also be like 120590119905= (1205902minus 1205901)28(1205902+ 1205901)

cos 2120573 = (1205901minus 1205902)2(1205902+ 1205901)

When (1205901+ 31205902) le 0 the fracture criterion is

1205902= minus120590119905

sin 2120573 = 0

(21)

where 120590119905is the tensile stress from the rock unidirectional

tensile test 120573 is the angle between the long-axis of theoval fracture and the principal compressive stress axis Thefracture angle is 120579 = 120573

If a new fracture occurs it will be in the direction of ovalfractureThe angle between the new fracture and the originalone is 120595 (Figure 11) The following equation can be obtainedby the fracture criterion

120595 = minus2120573 (22)

Equation (22) shows that the angle between the newfracture and the original one is 2120573 in the condition of (120590

1+

31205902) gt 0 The negative sign indicates the direction of the

new fracture is clockwise rotation by 120573 from the principalcompression stress axis (Figure 11)

For the three-dimensional problem Griffith fracturecriterion can be depicted as follows

When (1205901+ 31205903) gt 0 the fracture criterion is

(1205901minus 1205902)2+ (1205902minus 1205903)2+ (1205903minus 1205901)2

= 24120590119905(1205901+ 1205902+ 1205903)

cos 2120579 =1205901minus 1205903

2 (1205901+ 1205903)

(23)

When (1205901+ 31205903) le 0 the fracture criterion is

120590119905= minus1205903 (24)

Based on the fracture criterion above it can be judgedonly whether the tensile fracture or the shear fracture occurswhile the developing degrees of the fracture cannot bedetermined In fact with the asymmetry of the tectonic stressdistribution the fracture developing degrees of tectonic partdiffer The concept of fracture factor is introduced to judgethe fracture developing degrees of the rock quantitativelyThefracture factor is given by

119868120590=

100381610038161003816100381610038161003816100381610038161003816

120590119905

[120590119905]

100381610038161003816100381610038161003816100381610038161003816

or 119868119905=

100381610038161003816100381610038161003816100381610038161003816

120591119899

[120591119899]

100381610038161003816100381610038161003816100381610038161003816

(25)

Mathematical Problems in Engineering 9

568

568

281281

(a) 000m around the shaft wall

171

171

4343

(b) 060m around the shaft wall104

104

3737

(c) 105m around the shaft wall

85

85

5858

(d) 165m around the shaft wall

78

78

6767

(e) 255m around the shaft wall

Figure 9 von Mises stress distribution of the shaft wall in well UGS1

10 Mathematical Problems in Engineering

291

291

125125

(a) 00m around the shaft wall

89

89

1515

(b) 060m around the shaft wall

57

57

2323

(c) 105m around the shaft wall

47

47

3434

(d) 165m around the shaft wall

44

44

3838

(e) 255m around the shaft wall

Figure 10 The maximum shearing stress distribution around the shaft wall in UGS1

Mathematical Problems in Engineering 11

Table 5 Hole rock tensile fracture analysis results in UGS

Well Maximum principalstressMPa

VerticalstressMPa

Minimum principalstressMPa

PorepressureMPa

TensilestressMPa

Tensile strength[120590119905]MPa

Tensile fracturefactor 119868

120590

UGS6 minus2169 minus2597 minus2793 1200 095 497 0191UGS7 minus2132 minus2610 minus2602 1200 093 484 0192UGS8 minus2121 minus2503 minus2725 1200 099 492 0201UGS9 minus2066 minus2310 minus2783 1200 092 494 0186UGS10 minus1906 minus2515 minus2215 1200 098 495 0198

Table 6 Hole rock shear fracture analysis results in UGS

Well Maximum principalstressMPa

VerticalstressMPa

Minimum principalstressMPa

CohesivestressMPa

Maximum shearingstressMPa

Shear strength[120591119899]MPa

Shear fracturefactor 119868

119905

UGS6 minus2169 minus2597 minus2793 466 249 354 0703UGS7 minus2132 minus2610 minus2602 435 222 368 0603UGS8 minus2121 minus2503 minus2725 405 227 372 0510UGS9 minus2066 minus2310 minus2783 462 213 347 0614UGS10 minus1906 minus2515 minus2215 450 269 357 0754

The newfracture

The newfracture

x

y

120595 = minus2120573

120595 = minus2120573

120573

12059011205901

1205902

1205902

Figure 11 Fracture expansion direction of the oval fracture

where 119868120590and 119868119905are tensile fracture factor and shear fracture

factor 120590119905and 120591

119899are tensile stress and shearing stress in

the fracture plane [120590119905] and [120591

119899] are tensile strength and

shearing resistance The shaft rock will not be cracked whenthe fracture factor is below 1

72 Rock Fracture Analysis Results in the UGS According tothe Griffith fracture criterion presented above we conductthe tensile fracture analysis on the rock of the sidewall inUGS of a Turkey salt cavern With the obtained maximumand minimum horizontal in situ stresses of the test wells

r

120590h1

120590h2

120590h2

120590h1

120590120579120591r120579

120590r

Figure 12 The calculation model of wellbore stress

the tensile stress and shearing stress can be calculated bythe classical elastic theory The calculation model is shownin Figure 12 where 120590

ℎ1 120590ℎ2are the maximum and minimum

horizontal principal in situ stresses 120590119903is the radial normal

stress 120590120579is the tangential normal stress and 120591

119903120579is the

shearing stressThe results are shown in Table 5 showing that the

maximum tensile stress is smaller than the tensile strengthTherefore on the rock of the shaft wall no fracture will bedeveloped

Moreover according to the Coulomb-Navier failure cri-terion the shear fracture analysis is conducted on the rock ofthe shaft wall in UGS of a Turkey salt cavern Table 6 showsthe analysis results The shear fracture factors 119868

119905 in the hole

are all under 1 Given that themaximum shearing stress is lessthan the shearing strength the rock of the shaft wall will notdevelop a shear fracture

12 Mathematical Problems in Engineering

8 Conclusion

In this paper we improved the nonuniqueness of the in situstress field inversion solution by a priori constraints Fromthe perspective of basic equations of elastic mechanics anin situ stress numerical inversion model is built and basicfactors of in situ stress field are considered in applying thefinite element inversion method on the boundary loads ofthe in situ stress field An appropriate geometrical modeland suitable boundary conditions ensured an accurate in situstress field is derived The method is successfully applied inUGS of a Turkey salt cavern The optimisation results onprincipal stress are close to those from measurement By theoptimisation the overall in situ stress field of the researcharea is obtained and the stress distribution of the shaft wallis presented according to the calculation results The safetyof the shaft wall is confirmed by applying the correspondingfracture criterion

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to acknowledge the financial support ofthe National Natural Science Foundation of China (Grantsnos 51374228 51274231 and U1262208) and the CNPC KeyLaboratory Fund (Contract no 2014A-4214)

References

[1] B S Aadnoy ldquoInversion technique to determine the in-situstress field from fracturing datardquo Journal of Petroleum Scienceand Engineering vol 4 no 2 pp 127ndash141 1990

[2] M Bloch C A M Siqueira F H Ferreira and J C JConceicao ldquoTechniques for determining in-situ stress directionand magnitudesrdquo in Proceedings of the Latin American andCaribbean Petroleum Engineering Conference SPE 39075-MSRio de Janeiro Brazil August-September 1997

[3] J Huang D V Griffiths and S-W Wong ldquoIn situ stress deter-mination from inversion of hydraulic fracturing datardquo Interna-tional Journal of Rock Mechanics and Mining Sciences vol 48no 3 pp 476ndash481 2011

[4] J Djurhuus and B S Aadnoslashy ldquoIn situ stress state from inversionof fracturing data from oil wells and borehole image logsrdquoJournal of Petroleum Science and Engineering vol 38 no 3-4pp 121ndash130 2003

[5] B S Aadnoy ldquoIn-situ stress directions from borehole fracturetracesrdquo Journal of Petroleum Science and Engineering vol 4 no2 pp 143ndash153 1990

[6] C A Barton D Moos P Peska and M D Zoback ldquoUtilizingwellbore image data to determine the complete stress tensorapplication to permeability anisotropy and wellbore stabilityrdquoThe Log Analyst vol 38 no 6 pp 21ndash33 1997

[7] H Zhao F Ma J Xu and J Guo ldquoIn situ stress field inversionand its application in mining-induced rock mass movementrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 53 pp 120ndash128 2012

[8] Z D Chen Q A Meng and T F Wan ldquoNumerical simulationof tectonic stress field in Gulong depression in Songliao Basinusing elasitic-plastic increment methodrdquo Earth Science Fron-tiers vol 9 no 2 pp 483ndash491 2002 (Chinese)

[9] D K Singha and R Chatterjee ldquoGeomechanical modelingusing finite element method for prediction of in-situ stress inKrishna-Godavari basin Indiardquo International Journal of RockMechanics and Mining Sciences vol 73 no 5 pp 15ndash27 2015

[10] G Maier R Ardito and R Fedele ldquoInverse analysis problemsin structural engineering of concrete damsrdquo ComputationalMechanics vol 9 pp 5ndash10 2004

[11] G Maier M Bocciarelli G Bolzon and R Fedele ldquoInverseanalyses in fracture mechanicsrdquo International Journal of Frac-ture vol 138 no 1ndash4 pp 47ndash73 2006

[12] F Yin and D L Gao ldquoMechanical analysis and design of casingin directional well under in-situ stressesrdquo Journal of Natural GasScience and Engineering vol 20 pp 285ndash291 2014

[13] S Sakurai and S Akutagawa ldquoReport some aspects of backanalysis in geotechnical engineeringrdquo in Proceedings of theISRM International Symposium (EUROCK rsquo93) pp 1133ndash1140Lisboa Portugal June 1993

[14] L Z Xue Engineering Optimum Programming TechnologiesTianjing University Press Tianjing China 1998 (Chinese)

[15] F Li J-A Wang and J C Brigham ldquoInverse calculation of insitu stress in rock mass using the Surrogate-Model AcceleratedRandom Search algorithmrdquo Computers and Geotechnics vol 61pp 24ndash32 2014

[16] X Z Wang W Q Liu Y G Sun and Y J Ma ldquoField testingof fracturing techanolgy in coal formed gas wellrdquoOil Drilling ampProduction Technology vol 23 no 2 pp 58ndash61 2001 (Chinese)

[17] J Teng Y B Yao D M Liu and Y D Cai ldquoEvaluation ofcoal texture distributions in the southern Qinshui basin NorthChina investigation by amultiple geophysical loggingmethodrdquoInternational Journal of Coal Geology vol 140 pp 9ndash22 2015

[18] Z Meng J Zhang and R Wang ldquoIn-situ stress pore pressureand stress-dependent permeability in the Southern QinshuiBasinrdquo International Journal of Rock Mechanics and MiningSciences vol 48 no 1 pp 122ndash131 2011

[19] P Lancaster and K Salkauskas ldquoSurfaces generated by movingleast squaresmethodsrdquoMathematics of Computation vol 37 no155 pp 141ndash158 1981

[20] AHMedley ldquoStorage of natural gas in salt cavernsrdquo in Proceed-ings of the SPE Northern Plains Section Regional Meeting SPE-5427-MS Conference Paper Society of Petroleum EngineersOmaha Neb USA May 1975

[21] G Han M S Bruno K Lao J Young and L Dorfmann ldquoGasstorage and operations in single-bedded salt caverns stabilityanalysesrdquo SPEProduction andOperations vol 22 no 3 pp 368ndash376 2007

[22] P Berest and B Brouard ldquoLong-term behavior of salt cavernsrdquoin Proceedings of the 48th US Rock MechanicsGeomechanicsSymposium vol 14 Minneapolis Minn USA June 2014ARMA-2014-7032

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Table 4 Optimum results of the boundary load unit MPa

1198751

1198752

1198753

1198754

1198755

1198756

1198757

1198758

1198759

11987510

11987511

11987512

2639 1999 75277 2113 2275 9216 5328 12053 1484 1021 5257 2977

2000

4000

6000

8000

10000

12000

14000

4000 6000 8000 100002000

minus24

minus38

minus43

minus43

minus43

minus38

minus43

minus43

minus33

minus38

minus28

minus33

minus38minus43

minus25

minus26

minus27

minus28

minus29

minus30

minus31

minus32

minus33

minus34

minus35

minus36

minus37

minus38

minus39

minus40

minus41

minus42

minus43

minus44

minus45

minus46

minus47

minus48

Figure 8 Distribution of the minimum principal stress and orien-tation

As shown in Figure 10 just like the von Mises stress theshearing stress distribution is in an oval shape near the shaftwall and changes to a butterfly-like shape from the shaft wallThe shearing stress on the shaft wall is maximum (291MPa)due to the stress concentration and it decreases rapidly awayfrom the shaft wall with 89MPa when 06m away Shearingstress tends to be stable when it is 16m away from the shaftwall

From the stress analysis above we can draw the conclu-sion as the following

The stress concentration phenomenon does exist aroundthe hole The von Mises stress and shearing stress aremaximum on the shaft wall The stress distribution is in anoval shape around the shaft wall while it is in a butterfly-likeshape near the hole Both stresses decrease rapidly when theyare 1m away from the shaft wall and tend to be stable when13m away from the shaft wall

7 The Hole Stability Analysis of UGS

71 The Rock Fracture Analysis around the Hole of UGSBased on the Griffith Principle Griffith defined the inherentdefects or micro-fracture in the material as the Griffithfracture Based on theGriffith criterion [22] the rock fracturecondition in the planar stress state is shown below

When (1205901+ 31205902) gt 0 the fracture criterion is

(1205902minus 1205901)2= 8 (120590

2+ 1205901) 120590119905 (20)

Equation (20) can also be like 120590119905= (1205902minus 1205901)28(1205902+ 1205901)

cos 2120573 = (1205901minus 1205902)2(1205902+ 1205901)

When (1205901+ 31205902) le 0 the fracture criterion is

1205902= minus120590119905

sin 2120573 = 0

(21)

where 120590119905is the tensile stress from the rock unidirectional

tensile test 120573 is the angle between the long-axis of theoval fracture and the principal compressive stress axis Thefracture angle is 120579 = 120573

If a new fracture occurs it will be in the direction of ovalfractureThe angle between the new fracture and the originalone is 120595 (Figure 11) The following equation can be obtainedby the fracture criterion

120595 = minus2120573 (22)

Equation (22) shows that the angle between the newfracture and the original one is 2120573 in the condition of (120590

1+

31205902) gt 0 The negative sign indicates the direction of the

new fracture is clockwise rotation by 120573 from the principalcompression stress axis (Figure 11)

For the three-dimensional problem Griffith fracturecriterion can be depicted as follows

When (1205901+ 31205903) gt 0 the fracture criterion is

(1205901minus 1205902)2+ (1205902minus 1205903)2+ (1205903minus 1205901)2

= 24120590119905(1205901+ 1205902+ 1205903)

cos 2120579 =1205901minus 1205903

2 (1205901+ 1205903)

(23)

When (1205901+ 31205903) le 0 the fracture criterion is

120590119905= minus1205903 (24)

Based on the fracture criterion above it can be judgedonly whether the tensile fracture or the shear fracture occurswhile the developing degrees of the fracture cannot bedetermined In fact with the asymmetry of the tectonic stressdistribution the fracture developing degrees of tectonic partdiffer The concept of fracture factor is introduced to judgethe fracture developing degrees of the rock quantitativelyThefracture factor is given by

119868120590=

100381610038161003816100381610038161003816100381610038161003816

120590119905

[120590119905]

100381610038161003816100381610038161003816100381610038161003816

or 119868119905=

100381610038161003816100381610038161003816100381610038161003816

120591119899

[120591119899]

100381610038161003816100381610038161003816100381610038161003816

(25)

Mathematical Problems in Engineering 9

568

568

281281

(a) 000m around the shaft wall

171

171

4343

(b) 060m around the shaft wall104

104

3737

(c) 105m around the shaft wall

85

85

5858

(d) 165m around the shaft wall

78

78

6767

(e) 255m around the shaft wall

Figure 9 von Mises stress distribution of the shaft wall in well UGS1

10 Mathematical Problems in Engineering

291

291

125125

(a) 00m around the shaft wall

89

89

1515

(b) 060m around the shaft wall

57

57

2323

(c) 105m around the shaft wall

47

47

3434

(d) 165m around the shaft wall

44

44

3838

(e) 255m around the shaft wall

Figure 10 The maximum shearing stress distribution around the shaft wall in UGS1

Mathematical Problems in Engineering 11

Table 5 Hole rock tensile fracture analysis results in UGS

Well Maximum principalstressMPa

VerticalstressMPa

Minimum principalstressMPa

PorepressureMPa

TensilestressMPa

Tensile strength[120590119905]MPa

Tensile fracturefactor 119868

120590

UGS6 minus2169 minus2597 minus2793 1200 095 497 0191UGS7 minus2132 minus2610 minus2602 1200 093 484 0192UGS8 minus2121 minus2503 minus2725 1200 099 492 0201UGS9 minus2066 minus2310 minus2783 1200 092 494 0186UGS10 minus1906 minus2515 minus2215 1200 098 495 0198

Table 6 Hole rock shear fracture analysis results in UGS

Well Maximum principalstressMPa

VerticalstressMPa

Minimum principalstressMPa

CohesivestressMPa

Maximum shearingstressMPa

Shear strength[120591119899]MPa

Shear fracturefactor 119868

119905

UGS6 minus2169 minus2597 minus2793 466 249 354 0703UGS7 minus2132 minus2610 minus2602 435 222 368 0603UGS8 minus2121 minus2503 minus2725 405 227 372 0510UGS9 minus2066 minus2310 minus2783 462 213 347 0614UGS10 minus1906 minus2515 minus2215 450 269 357 0754

The newfracture

The newfracture

x

y

120595 = minus2120573

120595 = minus2120573

120573

12059011205901

1205902

1205902

Figure 11 Fracture expansion direction of the oval fracture

where 119868120590and 119868119905are tensile fracture factor and shear fracture

factor 120590119905and 120591

119899are tensile stress and shearing stress in

the fracture plane [120590119905] and [120591

119899] are tensile strength and

shearing resistance The shaft rock will not be cracked whenthe fracture factor is below 1

72 Rock Fracture Analysis Results in the UGS According tothe Griffith fracture criterion presented above we conductthe tensile fracture analysis on the rock of the sidewall inUGS of a Turkey salt cavern With the obtained maximumand minimum horizontal in situ stresses of the test wells

r

120590h1

120590h2

120590h2

120590h1

120590120579120591r120579

120590r

Figure 12 The calculation model of wellbore stress

the tensile stress and shearing stress can be calculated bythe classical elastic theory The calculation model is shownin Figure 12 where 120590

ℎ1 120590ℎ2are the maximum and minimum

horizontal principal in situ stresses 120590119903is the radial normal

stress 120590120579is the tangential normal stress and 120591

119903120579is the

shearing stressThe results are shown in Table 5 showing that the

maximum tensile stress is smaller than the tensile strengthTherefore on the rock of the shaft wall no fracture will bedeveloped

Moreover according to the Coulomb-Navier failure cri-terion the shear fracture analysis is conducted on the rock ofthe shaft wall in UGS of a Turkey salt cavern Table 6 showsthe analysis results The shear fracture factors 119868

119905 in the hole

are all under 1 Given that themaximum shearing stress is lessthan the shearing strength the rock of the shaft wall will notdevelop a shear fracture

12 Mathematical Problems in Engineering

8 Conclusion

In this paper we improved the nonuniqueness of the in situstress field inversion solution by a priori constraints Fromthe perspective of basic equations of elastic mechanics anin situ stress numerical inversion model is built and basicfactors of in situ stress field are considered in applying thefinite element inversion method on the boundary loads ofthe in situ stress field An appropriate geometrical modeland suitable boundary conditions ensured an accurate in situstress field is derived The method is successfully applied inUGS of a Turkey salt cavern The optimisation results onprincipal stress are close to those from measurement By theoptimisation the overall in situ stress field of the researcharea is obtained and the stress distribution of the shaft wallis presented according to the calculation results The safetyof the shaft wall is confirmed by applying the correspondingfracture criterion

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to acknowledge the financial support ofthe National Natural Science Foundation of China (Grantsnos 51374228 51274231 and U1262208) and the CNPC KeyLaboratory Fund (Contract no 2014A-4214)

References

[1] B S Aadnoy ldquoInversion technique to determine the in-situstress field from fracturing datardquo Journal of Petroleum Scienceand Engineering vol 4 no 2 pp 127ndash141 1990

[2] M Bloch C A M Siqueira F H Ferreira and J C JConceicao ldquoTechniques for determining in-situ stress directionand magnitudesrdquo in Proceedings of the Latin American andCaribbean Petroleum Engineering Conference SPE 39075-MSRio de Janeiro Brazil August-September 1997

[3] J Huang D V Griffiths and S-W Wong ldquoIn situ stress deter-mination from inversion of hydraulic fracturing datardquo Interna-tional Journal of Rock Mechanics and Mining Sciences vol 48no 3 pp 476ndash481 2011

[4] J Djurhuus and B S Aadnoslashy ldquoIn situ stress state from inversionof fracturing data from oil wells and borehole image logsrdquoJournal of Petroleum Science and Engineering vol 38 no 3-4pp 121ndash130 2003

[5] B S Aadnoy ldquoIn-situ stress directions from borehole fracturetracesrdquo Journal of Petroleum Science and Engineering vol 4 no2 pp 143ndash153 1990

[6] C A Barton D Moos P Peska and M D Zoback ldquoUtilizingwellbore image data to determine the complete stress tensorapplication to permeability anisotropy and wellbore stabilityrdquoThe Log Analyst vol 38 no 6 pp 21ndash33 1997

[7] H Zhao F Ma J Xu and J Guo ldquoIn situ stress field inversionand its application in mining-induced rock mass movementrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 53 pp 120ndash128 2012

[8] Z D Chen Q A Meng and T F Wan ldquoNumerical simulationof tectonic stress field in Gulong depression in Songliao Basinusing elasitic-plastic increment methodrdquo Earth Science Fron-tiers vol 9 no 2 pp 483ndash491 2002 (Chinese)

[9] D K Singha and R Chatterjee ldquoGeomechanical modelingusing finite element method for prediction of in-situ stress inKrishna-Godavari basin Indiardquo International Journal of RockMechanics and Mining Sciences vol 73 no 5 pp 15ndash27 2015

[10] G Maier R Ardito and R Fedele ldquoInverse analysis problemsin structural engineering of concrete damsrdquo ComputationalMechanics vol 9 pp 5ndash10 2004

[11] G Maier M Bocciarelli G Bolzon and R Fedele ldquoInverseanalyses in fracture mechanicsrdquo International Journal of Frac-ture vol 138 no 1ndash4 pp 47ndash73 2006

[12] F Yin and D L Gao ldquoMechanical analysis and design of casingin directional well under in-situ stressesrdquo Journal of Natural GasScience and Engineering vol 20 pp 285ndash291 2014

[13] S Sakurai and S Akutagawa ldquoReport some aspects of backanalysis in geotechnical engineeringrdquo in Proceedings of theISRM International Symposium (EUROCK rsquo93) pp 1133ndash1140Lisboa Portugal June 1993

[14] L Z Xue Engineering Optimum Programming TechnologiesTianjing University Press Tianjing China 1998 (Chinese)

[15] F Li J-A Wang and J C Brigham ldquoInverse calculation of insitu stress in rock mass using the Surrogate-Model AcceleratedRandom Search algorithmrdquo Computers and Geotechnics vol 61pp 24ndash32 2014

[16] X Z Wang W Q Liu Y G Sun and Y J Ma ldquoField testingof fracturing techanolgy in coal formed gas wellrdquoOil Drilling ampProduction Technology vol 23 no 2 pp 58ndash61 2001 (Chinese)

[17] J Teng Y B Yao D M Liu and Y D Cai ldquoEvaluation ofcoal texture distributions in the southern Qinshui basin NorthChina investigation by amultiple geophysical loggingmethodrdquoInternational Journal of Coal Geology vol 140 pp 9ndash22 2015

[18] Z Meng J Zhang and R Wang ldquoIn-situ stress pore pressureand stress-dependent permeability in the Southern QinshuiBasinrdquo International Journal of Rock Mechanics and MiningSciences vol 48 no 1 pp 122ndash131 2011

[19] P Lancaster and K Salkauskas ldquoSurfaces generated by movingleast squaresmethodsrdquoMathematics of Computation vol 37 no155 pp 141ndash158 1981

[20] AHMedley ldquoStorage of natural gas in salt cavernsrdquo in Proceed-ings of the SPE Northern Plains Section Regional Meeting SPE-5427-MS Conference Paper Society of Petroleum EngineersOmaha Neb USA May 1975

[21] G Han M S Bruno K Lao J Young and L Dorfmann ldquoGasstorage and operations in single-bedded salt caverns stabilityanalysesrdquo SPEProduction andOperations vol 22 no 3 pp 368ndash376 2007

[22] P Berest and B Brouard ldquoLong-term behavior of salt cavernsrdquoin Proceedings of the 48th US Rock MechanicsGeomechanicsSymposium vol 14 Minneapolis Minn USA June 2014ARMA-2014-7032

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

568

568

281281

(a) 000m around the shaft wall

171

171

4343

(b) 060m around the shaft wall104

104

3737

(c) 105m around the shaft wall

85

85

5858

(d) 165m around the shaft wall

78

78

6767

(e) 255m around the shaft wall

Figure 9 von Mises stress distribution of the shaft wall in well UGS1

10 Mathematical Problems in Engineering

291

291

125125

(a) 00m around the shaft wall

89

89

1515

(b) 060m around the shaft wall

57

57

2323

(c) 105m around the shaft wall

47

47

3434

(d) 165m around the shaft wall

44

44

3838

(e) 255m around the shaft wall

Figure 10 The maximum shearing stress distribution around the shaft wall in UGS1

Mathematical Problems in Engineering 11

Table 5 Hole rock tensile fracture analysis results in UGS

Well Maximum principalstressMPa

VerticalstressMPa

Minimum principalstressMPa

PorepressureMPa

TensilestressMPa

Tensile strength[120590119905]MPa

Tensile fracturefactor 119868

120590

UGS6 minus2169 minus2597 minus2793 1200 095 497 0191UGS7 minus2132 minus2610 minus2602 1200 093 484 0192UGS8 minus2121 minus2503 minus2725 1200 099 492 0201UGS9 minus2066 minus2310 minus2783 1200 092 494 0186UGS10 minus1906 minus2515 minus2215 1200 098 495 0198

Table 6 Hole rock shear fracture analysis results in UGS

Well Maximum principalstressMPa

VerticalstressMPa

Minimum principalstressMPa

CohesivestressMPa

Maximum shearingstressMPa

Shear strength[120591119899]MPa

Shear fracturefactor 119868

119905

UGS6 minus2169 minus2597 minus2793 466 249 354 0703UGS7 minus2132 minus2610 minus2602 435 222 368 0603UGS8 minus2121 minus2503 minus2725 405 227 372 0510UGS9 minus2066 minus2310 minus2783 462 213 347 0614UGS10 minus1906 minus2515 minus2215 450 269 357 0754

The newfracture

The newfracture

x

y

120595 = minus2120573

120595 = minus2120573

120573

12059011205901

1205902

1205902

Figure 11 Fracture expansion direction of the oval fracture

where 119868120590and 119868119905are tensile fracture factor and shear fracture

factor 120590119905and 120591

119899are tensile stress and shearing stress in

the fracture plane [120590119905] and [120591

119899] are tensile strength and

shearing resistance The shaft rock will not be cracked whenthe fracture factor is below 1

72 Rock Fracture Analysis Results in the UGS According tothe Griffith fracture criterion presented above we conductthe tensile fracture analysis on the rock of the sidewall inUGS of a Turkey salt cavern With the obtained maximumand minimum horizontal in situ stresses of the test wells

r

120590h1

120590h2

120590h2

120590h1

120590120579120591r120579

120590r

Figure 12 The calculation model of wellbore stress

the tensile stress and shearing stress can be calculated bythe classical elastic theory The calculation model is shownin Figure 12 where 120590

ℎ1 120590ℎ2are the maximum and minimum

horizontal principal in situ stresses 120590119903is the radial normal

stress 120590120579is the tangential normal stress and 120591

119903120579is the

shearing stressThe results are shown in Table 5 showing that the

maximum tensile stress is smaller than the tensile strengthTherefore on the rock of the shaft wall no fracture will bedeveloped

Moreover according to the Coulomb-Navier failure cri-terion the shear fracture analysis is conducted on the rock ofthe shaft wall in UGS of a Turkey salt cavern Table 6 showsthe analysis results The shear fracture factors 119868

119905 in the hole

are all under 1 Given that themaximum shearing stress is lessthan the shearing strength the rock of the shaft wall will notdevelop a shear fracture

12 Mathematical Problems in Engineering

8 Conclusion

In this paper we improved the nonuniqueness of the in situstress field inversion solution by a priori constraints Fromthe perspective of basic equations of elastic mechanics anin situ stress numerical inversion model is built and basicfactors of in situ stress field are considered in applying thefinite element inversion method on the boundary loads ofthe in situ stress field An appropriate geometrical modeland suitable boundary conditions ensured an accurate in situstress field is derived The method is successfully applied inUGS of a Turkey salt cavern The optimisation results onprincipal stress are close to those from measurement By theoptimisation the overall in situ stress field of the researcharea is obtained and the stress distribution of the shaft wallis presented according to the calculation results The safetyof the shaft wall is confirmed by applying the correspondingfracture criterion

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to acknowledge the financial support ofthe National Natural Science Foundation of China (Grantsnos 51374228 51274231 and U1262208) and the CNPC KeyLaboratory Fund (Contract no 2014A-4214)

References

[1] B S Aadnoy ldquoInversion technique to determine the in-situstress field from fracturing datardquo Journal of Petroleum Scienceand Engineering vol 4 no 2 pp 127ndash141 1990

[2] M Bloch C A M Siqueira F H Ferreira and J C JConceicao ldquoTechniques for determining in-situ stress directionand magnitudesrdquo in Proceedings of the Latin American andCaribbean Petroleum Engineering Conference SPE 39075-MSRio de Janeiro Brazil August-September 1997

[3] J Huang D V Griffiths and S-W Wong ldquoIn situ stress deter-mination from inversion of hydraulic fracturing datardquo Interna-tional Journal of Rock Mechanics and Mining Sciences vol 48no 3 pp 476ndash481 2011

[4] J Djurhuus and B S Aadnoslashy ldquoIn situ stress state from inversionof fracturing data from oil wells and borehole image logsrdquoJournal of Petroleum Science and Engineering vol 38 no 3-4pp 121ndash130 2003

[5] B S Aadnoy ldquoIn-situ stress directions from borehole fracturetracesrdquo Journal of Petroleum Science and Engineering vol 4 no2 pp 143ndash153 1990

[6] C A Barton D Moos P Peska and M D Zoback ldquoUtilizingwellbore image data to determine the complete stress tensorapplication to permeability anisotropy and wellbore stabilityrdquoThe Log Analyst vol 38 no 6 pp 21ndash33 1997

[7] H Zhao F Ma J Xu and J Guo ldquoIn situ stress field inversionand its application in mining-induced rock mass movementrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 53 pp 120ndash128 2012

[8] Z D Chen Q A Meng and T F Wan ldquoNumerical simulationof tectonic stress field in Gulong depression in Songliao Basinusing elasitic-plastic increment methodrdquo Earth Science Fron-tiers vol 9 no 2 pp 483ndash491 2002 (Chinese)

[9] D K Singha and R Chatterjee ldquoGeomechanical modelingusing finite element method for prediction of in-situ stress inKrishna-Godavari basin Indiardquo International Journal of RockMechanics and Mining Sciences vol 73 no 5 pp 15ndash27 2015

[10] G Maier R Ardito and R Fedele ldquoInverse analysis problemsin structural engineering of concrete damsrdquo ComputationalMechanics vol 9 pp 5ndash10 2004

[11] G Maier M Bocciarelli G Bolzon and R Fedele ldquoInverseanalyses in fracture mechanicsrdquo International Journal of Frac-ture vol 138 no 1ndash4 pp 47ndash73 2006

[12] F Yin and D L Gao ldquoMechanical analysis and design of casingin directional well under in-situ stressesrdquo Journal of Natural GasScience and Engineering vol 20 pp 285ndash291 2014

[13] S Sakurai and S Akutagawa ldquoReport some aspects of backanalysis in geotechnical engineeringrdquo in Proceedings of theISRM International Symposium (EUROCK rsquo93) pp 1133ndash1140Lisboa Portugal June 1993

[14] L Z Xue Engineering Optimum Programming TechnologiesTianjing University Press Tianjing China 1998 (Chinese)

[15] F Li J-A Wang and J C Brigham ldquoInverse calculation of insitu stress in rock mass using the Surrogate-Model AcceleratedRandom Search algorithmrdquo Computers and Geotechnics vol 61pp 24ndash32 2014

[16] X Z Wang W Q Liu Y G Sun and Y J Ma ldquoField testingof fracturing techanolgy in coal formed gas wellrdquoOil Drilling ampProduction Technology vol 23 no 2 pp 58ndash61 2001 (Chinese)

[17] J Teng Y B Yao D M Liu and Y D Cai ldquoEvaluation ofcoal texture distributions in the southern Qinshui basin NorthChina investigation by amultiple geophysical loggingmethodrdquoInternational Journal of Coal Geology vol 140 pp 9ndash22 2015

[18] Z Meng J Zhang and R Wang ldquoIn-situ stress pore pressureand stress-dependent permeability in the Southern QinshuiBasinrdquo International Journal of Rock Mechanics and MiningSciences vol 48 no 1 pp 122ndash131 2011

[19] P Lancaster and K Salkauskas ldquoSurfaces generated by movingleast squaresmethodsrdquoMathematics of Computation vol 37 no155 pp 141ndash158 1981

[20] AHMedley ldquoStorage of natural gas in salt cavernsrdquo in Proceed-ings of the SPE Northern Plains Section Regional Meeting SPE-5427-MS Conference Paper Society of Petroleum EngineersOmaha Neb USA May 1975

[21] G Han M S Bruno K Lao J Young and L Dorfmann ldquoGasstorage and operations in single-bedded salt caverns stabilityanalysesrdquo SPEProduction andOperations vol 22 no 3 pp 368ndash376 2007

[22] P Berest and B Brouard ldquoLong-term behavior of salt cavernsrdquoin Proceedings of the 48th US Rock MechanicsGeomechanicsSymposium vol 14 Minneapolis Minn USA June 2014ARMA-2014-7032

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

291

291

125125

(a) 00m around the shaft wall

89

89

1515

(b) 060m around the shaft wall

57

57

2323

(c) 105m around the shaft wall

47

47

3434

(d) 165m around the shaft wall

44

44

3838

(e) 255m around the shaft wall

Figure 10 The maximum shearing stress distribution around the shaft wall in UGS1

Mathematical Problems in Engineering 11

Table 5 Hole rock tensile fracture analysis results in UGS

Well Maximum principalstressMPa

VerticalstressMPa

Minimum principalstressMPa

PorepressureMPa

TensilestressMPa

Tensile strength[120590119905]MPa

Tensile fracturefactor 119868

120590

UGS6 minus2169 minus2597 minus2793 1200 095 497 0191UGS7 minus2132 minus2610 minus2602 1200 093 484 0192UGS8 minus2121 minus2503 minus2725 1200 099 492 0201UGS9 minus2066 minus2310 minus2783 1200 092 494 0186UGS10 minus1906 minus2515 minus2215 1200 098 495 0198

Table 6 Hole rock shear fracture analysis results in UGS

Well Maximum principalstressMPa

VerticalstressMPa

Minimum principalstressMPa

CohesivestressMPa

Maximum shearingstressMPa

Shear strength[120591119899]MPa

Shear fracturefactor 119868

119905

UGS6 minus2169 minus2597 minus2793 466 249 354 0703UGS7 minus2132 minus2610 minus2602 435 222 368 0603UGS8 minus2121 minus2503 minus2725 405 227 372 0510UGS9 minus2066 minus2310 minus2783 462 213 347 0614UGS10 minus1906 minus2515 minus2215 450 269 357 0754

The newfracture

The newfracture

x

y

120595 = minus2120573

120595 = minus2120573

120573

12059011205901

1205902

1205902

Figure 11 Fracture expansion direction of the oval fracture

where 119868120590and 119868119905are tensile fracture factor and shear fracture

factor 120590119905and 120591

119899are tensile stress and shearing stress in

the fracture plane [120590119905] and [120591

119899] are tensile strength and

shearing resistance The shaft rock will not be cracked whenthe fracture factor is below 1

72 Rock Fracture Analysis Results in the UGS According tothe Griffith fracture criterion presented above we conductthe tensile fracture analysis on the rock of the sidewall inUGS of a Turkey salt cavern With the obtained maximumand minimum horizontal in situ stresses of the test wells

r

120590h1

120590h2

120590h2

120590h1

120590120579120591r120579

120590r

Figure 12 The calculation model of wellbore stress

the tensile stress and shearing stress can be calculated bythe classical elastic theory The calculation model is shownin Figure 12 where 120590

ℎ1 120590ℎ2are the maximum and minimum

horizontal principal in situ stresses 120590119903is the radial normal

stress 120590120579is the tangential normal stress and 120591

119903120579is the

shearing stressThe results are shown in Table 5 showing that the

maximum tensile stress is smaller than the tensile strengthTherefore on the rock of the shaft wall no fracture will bedeveloped

Moreover according to the Coulomb-Navier failure cri-terion the shear fracture analysis is conducted on the rock ofthe shaft wall in UGS of a Turkey salt cavern Table 6 showsthe analysis results The shear fracture factors 119868

119905 in the hole

are all under 1 Given that themaximum shearing stress is lessthan the shearing strength the rock of the shaft wall will notdevelop a shear fracture

12 Mathematical Problems in Engineering

8 Conclusion

In this paper we improved the nonuniqueness of the in situstress field inversion solution by a priori constraints Fromthe perspective of basic equations of elastic mechanics anin situ stress numerical inversion model is built and basicfactors of in situ stress field are considered in applying thefinite element inversion method on the boundary loads ofthe in situ stress field An appropriate geometrical modeland suitable boundary conditions ensured an accurate in situstress field is derived The method is successfully applied inUGS of a Turkey salt cavern The optimisation results onprincipal stress are close to those from measurement By theoptimisation the overall in situ stress field of the researcharea is obtained and the stress distribution of the shaft wallis presented according to the calculation results The safetyof the shaft wall is confirmed by applying the correspondingfracture criterion

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to acknowledge the financial support ofthe National Natural Science Foundation of China (Grantsnos 51374228 51274231 and U1262208) and the CNPC KeyLaboratory Fund (Contract no 2014A-4214)

References

[1] B S Aadnoy ldquoInversion technique to determine the in-situstress field from fracturing datardquo Journal of Petroleum Scienceand Engineering vol 4 no 2 pp 127ndash141 1990

[2] M Bloch C A M Siqueira F H Ferreira and J C JConceicao ldquoTechniques for determining in-situ stress directionand magnitudesrdquo in Proceedings of the Latin American andCaribbean Petroleum Engineering Conference SPE 39075-MSRio de Janeiro Brazil August-September 1997

[3] J Huang D V Griffiths and S-W Wong ldquoIn situ stress deter-mination from inversion of hydraulic fracturing datardquo Interna-tional Journal of Rock Mechanics and Mining Sciences vol 48no 3 pp 476ndash481 2011

[4] J Djurhuus and B S Aadnoslashy ldquoIn situ stress state from inversionof fracturing data from oil wells and borehole image logsrdquoJournal of Petroleum Science and Engineering vol 38 no 3-4pp 121ndash130 2003

[5] B S Aadnoy ldquoIn-situ stress directions from borehole fracturetracesrdquo Journal of Petroleum Science and Engineering vol 4 no2 pp 143ndash153 1990

[6] C A Barton D Moos P Peska and M D Zoback ldquoUtilizingwellbore image data to determine the complete stress tensorapplication to permeability anisotropy and wellbore stabilityrdquoThe Log Analyst vol 38 no 6 pp 21ndash33 1997

[7] H Zhao F Ma J Xu and J Guo ldquoIn situ stress field inversionand its application in mining-induced rock mass movementrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 53 pp 120ndash128 2012

[8] Z D Chen Q A Meng and T F Wan ldquoNumerical simulationof tectonic stress field in Gulong depression in Songliao Basinusing elasitic-plastic increment methodrdquo Earth Science Fron-tiers vol 9 no 2 pp 483ndash491 2002 (Chinese)

[9] D K Singha and R Chatterjee ldquoGeomechanical modelingusing finite element method for prediction of in-situ stress inKrishna-Godavari basin Indiardquo International Journal of RockMechanics and Mining Sciences vol 73 no 5 pp 15ndash27 2015

[10] G Maier R Ardito and R Fedele ldquoInverse analysis problemsin structural engineering of concrete damsrdquo ComputationalMechanics vol 9 pp 5ndash10 2004

[11] G Maier M Bocciarelli G Bolzon and R Fedele ldquoInverseanalyses in fracture mechanicsrdquo International Journal of Frac-ture vol 138 no 1ndash4 pp 47ndash73 2006

[12] F Yin and D L Gao ldquoMechanical analysis and design of casingin directional well under in-situ stressesrdquo Journal of Natural GasScience and Engineering vol 20 pp 285ndash291 2014

[13] S Sakurai and S Akutagawa ldquoReport some aspects of backanalysis in geotechnical engineeringrdquo in Proceedings of theISRM International Symposium (EUROCK rsquo93) pp 1133ndash1140Lisboa Portugal June 1993

[14] L Z Xue Engineering Optimum Programming TechnologiesTianjing University Press Tianjing China 1998 (Chinese)

[15] F Li J-A Wang and J C Brigham ldquoInverse calculation of insitu stress in rock mass using the Surrogate-Model AcceleratedRandom Search algorithmrdquo Computers and Geotechnics vol 61pp 24ndash32 2014

[16] X Z Wang W Q Liu Y G Sun and Y J Ma ldquoField testingof fracturing techanolgy in coal formed gas wellrdquoOil Drilling ampProduction Technology vol 23 no 2 pp 58ndash61 2001 (Chinese)

[17] J Teng Y B Yao D M Liu and Y D Cai ldquoEvaluation ofcoal texture distributions in the southern Qinshui basin NorthChina investigation by amultiple geophysical loggingmethodrdquoInternational Journal of Coal Geology vol 140 pp 9ndash22 2015

[18] Z Meng J Zhang and R Wang ldquoIn-situ stress pore pressureand stress-dependent permeability in the Southern QinshuiBasinrdquo International Journal of Rock Mechanics and MiningSciences vol 48 no 1 pp 122ndash131 2011

[19] P Lancaster and K Salkauskas ldquoSurfaces generated by movingleast squaresmethodsrdquoMathematics of Computation vol 37 no155 pp 141ndash158 1981

[20] AHMedley ldquoStorage of natural gas in salt cavernsrdquo in Proceed-ings of the SPE Northern Plains Section Regional Meeting SPE-5427-MS Conference Paper Society of Petroleum EngineersOmaha Neb USA May 1975

[21] G Han M S Bruno K Lao J Young and L Dorfmann ldquoGasstorage and operations in single-bedded salt caverns stabilityanalysesrdquo SPEProduction andOperations vol 22 no 3 pp 368ndash376 2007

[22] P Berest and B Brouard ldquoLong-term behavior of salt cavernsrdquoin Proceedings of the 48th US Rock MechanicsGeomechanicsSymposium vol 14 Minneapolis Minn USA June 2014ARMA-2014-7032

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

Table 5 Hole rock tensile fracture analysis results in UGS

Well Maximum principalstressMPa

VerticalstressMPa

Minimum principalstressMPa

PorepressureMPa

TensilestressMPa

Tensile strength[120590119905]MPa

Tensile fracturefactor 119868

120590

UGS6 minus2169 minus2597 minus2793 1200 095 497 0191UGS7 minus2132 minus2610 minus2602 1200 093 484 0192UGS8 minus2121 minus2503 minus2725 1200 099 492 0201UGS9 minus2066 minus2310 minus2783 1200 092 494 0186UGS10 minus1906 minus2515 minus2215 1200 098 495 0198

Table 6 Hole rock shear fracture analysis results in UGS

Well Maximum principalstressMPa

VerticalstressMPa

Minimum principalstressMPa

CohesivestressMPa

Maximum shearingstressMPa

Shear strength[120591119899]MPa

Shear fracturefactor 119868

119905

UGS6 minus2169 minus2597 minus2793 466 249 354 0703UGS7 minus2132 minus2610 minus2602 435 222 368 0603UGS8 minus2121 minus2503 minus2725 405 227 372 0510UGS9 minus2066 minus2310 minus2783 462 213 347 0614UGS10 minus1906 minus2515 minus2215 450 269 357 0754

The newfracture

The newfracture

x

y

120595 = minus2120573

120595 = minus2120573

120573

12059011205901

1205902

1205902

Figure 11 Fracture expansion direction of the oval fracture

where 119868120590and 119868119905are tensile fracture factor and shear fracture

factor 120590119905and 120591

119899are tensile stress and shearing stress in

the fracture plane [120590119905] and [120591

119899] are tensile strength and

shearing resistance The shaft rock will not be cracked whenthe fracture factor is below 1

72 Rock Fracture Analysis Results in the UGS According tothe Griffith fracture criterion presented above we conductthe tensile fracture analysis on the rock of the sidewall inUGS of a Turkey salt cavern With the obtained maximumand minimum horizontal in situ stresses of the test wells

r

120590h1

120590h2

120590h2

120590h1

120590120579120591r120579

120590r

Figure 12 The calculation model of wellbore stress

the tensile stress and shearing stress can be calculated bythe classical elastic theory The calculation model is shownin Figure 12 where 120590

ℎ1 120590ℎ2are the maximum and minimum

horizontal principal in situ stresses 120590119903is the radial normal

stress 120590120579is the tangential normal stress and 120591

119903120579is the

shearing stressThe results are shown in Table 5 showing that the

maximum tensile stress is smaller than the tensile strengthTherefore on the rock of the shaft wall no fracture will bedeveloped

Moreover according to the Coulomb-Navier failure cri-terion the shear fracture analysis is conducted on the rock ofthe shaft wall in UGS of a Turkey salt cavern Table 6 showsthe analysis results The shear fracture factors 119868

119905 in the hole

are all under 1 Given that themaximum shearing stress is lessthan the shearing strength the rock of the shaft wall will notdevelop a shear fracture

12 Mathematical Problems in Engineering

8 Conclusion

In this paper we improved the nonuniqueness of the in situstress field inversion solution by a priori constraints Fromthe perspective of basic equations of elastic mechanics anin situ stress numerical inversion model is built and basicfactors of in situ stress field are considered in applying thefinite element inversion method on the boundary loads ofthe in situ stress field An appropriate geometrical modeland suitable boundary conditions ensured an accurate in situstress field is derived The method is successfully applied inUGS of a Turkey salt cavern The optimisation results onprincipal stress are close to those from measurement By theoptimisation the overall in situ stress field of the researcharea is obtained and the stress distribution of the shaft wallis presented according to the calculation results The safetyof the shaft wall is confirmed by applying the correspondingfracture criterion

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to acknowledge the financial support ofthe National Natural Science Foundation of China (Grantsnos 51374228 51274231 and U1262208) and the CNPC KeyLaboratory Fund (Contract no 2014A-4214)

References

[1] B S Aadnoy ldquoInversion technique to determine the in-situstress field from fracturing datardquo Journal of Petroleum Scienceand Engineering vol 4 no 2 pp 127ndash141 1990

[2] M Bloch C A M Siqueira F H Ferreira and J C JConceicao ldquoTechniques for determining in-situ stress directionand magnitudesrdquo in Proceedings of the Latin American andCaribbean Petroleum Engineering Conference SPE 39075-MSRio de Janeiro Brazil August-September 1997

[3] J Huang D V Griffiths and S-W Wong ldquoIn situ stress deter-mination from inversion of hydraulic fracturing datardquo Interna-tional Journal of Rock Mechanics and Mining Sciences vol 48no 3 pp 476ndash481 2011

[4] J Djurhuus and B S Aadnoslashy ldquoIn situ stress state from inversionof fracturing data from oil wells and borehole image logsrdquoJournal of Petroleum Science and Engineering vol 38 no 3-4pp 121ndash130 2003

[5] B S Aadnoy ldquoIn-situ stress directions from borehole fracturetracesrdquo Journal of Petroleum Science and Engineering vol 4 no2 pp 143ndash153 1990

[6] C A Barton D Moos P Peska and M D Zoback ldquoUtilizingwellbore image data to determine the complete stress tensorapplication to permeability anisotropy and wellbore stabilityrdquoThe Log Analyst vol 38 no 6 pp 21ndash33 1997

[7] H Zhao F Ma J Xu and J Guo ldquoIn situ stress field inversionand its application in mining-induced rock mass movementrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 53 pp 120ndash128 2012

[8] Z D Chen Q A Meng and T F Wan ldquoNumerical simulationof tectonic stress field in Gulong depression in Songliao Basinusing elasitic-plastic increment methodrdquo Earth Science Fron-tiers vol 9 no 2 pp 483ndash491 2002 (Chinese)

[9] D K Singha and R Chatterjee ldquoGeomechanical modelingusing finite element method for prediction of in-situ stress inKrishna-Godavari basin Indiardquo International Journal of RockMechanics and Mining Sciences vol 73 no 5 pp 15ndash27 2015

[10] G Maier R Ardito and R Fedele ldquoInverse analysis problemsin structural engineering of concrete damsrdquo ComputationalMechanics vol 9 pp 5ndash10 2004

[11] G Maier M Bocciarelli G Bolzon and R Fedele ldquoInverseanalyses in fracture mechanicsrdquo International Journal of Frac-ture vol 138 no 1ndash4 pp 47ndash73 2006

[12] F Yin and D L Gao ldquoMechanical analysis and design of casingin directional well under in-situ stressesrdquo Journal of Natural GasScience and Engineering vol 20 pp 285ndash291 2014

[13] S Sakurai and S Akutagawa ldquoReport some aspects of backanalysis in geotechnical engineeringrdquo in Proceedings of theISRM International Symposium (EUROCK rsquo93) pp 1133ndash1140Lisboa Portugal June 1993

[14] L Z Xue Engineering Optimum Programming TechnologiesTianjing University Press Tianjing China 1998 (Chinese)

[15] F Li J-A Wang and J C Brigham ldquoInverse calculation of insitu stress in rock mass using the Surrogate-Model AcceleratedRandom Search algorithmrdquo Computers and Geotechnics vol 61pp 24ndash32 2014

[16] X Z Wang W Q Liu Y G Sun and Y J Ma ldquoField testingof fracturing techanolgy in coal formed gas wellrdquoOil Drilling ampProduction Technology vol 23 no 2 pp 58ndash61 2001 (Chinese)

[17] J Teng Y B Yao D M Liu and Y D Cai ldquoEvaluation ofcoal texture distributions in the southern Qinshui basin NorthChina investigation by amultiple geophysical loggingmethodrdquoInternational Journal of Coal Geology vol 140 pp 9ndash22 2015

[18] Z Meng J Zhang and R Wang ldquoIn-situ stress pore pressureand stress-dependent permeability in the Southern QinshuiBasinrdquo International Journal of Rock Mechanics and MiningSciences vol 48 no 1 pp 122ndash131 2011

[19] P Lancaster and K Salkauskas ldquoSurfaces generated by movingleast squaresmethodsrdquoMathematics of Computation vol 37 no155 pp 141ndash158 1981

[20] AHMedley ldquoStorage of natural gas in salt cavernsrdquo in Proceed-ings of the SPE Northern Plains Section Regional Meeting SPE-5427-MS Conference Paper Society of Petroleum EngineersOmaha Neb USA May 1975

[21] G Han M S Bruno K Lao J Young and L Dorfmann ldquoGasstorage and operations in single-bedded salt caverns stabilityanalysesrdquo SPEProduction andOperations vol 22 no 3 pp 368ndash376 2007

[22] P Berest and B Brouard ldquoLong-term behavior of salt cavernsrdquoin Proceedings of the 48th US Rock MechanicsGeomechanicsSymposium vol 14 Minneapolis Minn USA June 2014ARMA-2014-7032

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

8 Conclusion

In this paper we improved the nonuniqueness of the in situstress field inversion solution by a priori constraints Fromthe perspective of basic equations of elastic mechanics anin situ stress numerical inversion model is built and basicfactors of in situ stress field are considered in applying thefinite element inversion method on the boundary loads ofthe in situ stress field An appropriate geometrical modeland suitable boundary conditions ensured an accurate in situstress field is derived The method is successfully applied inUGS of a Turkey salt cavern The optimisation results onprincipal stress are close to those from measurement By theoptimisation the overall in situ stress field of the researcharea is obtained and the stress distribution of the shaft wallis presented according to the calculation results The safetyof the shaft wall is confirmed by applying the correspondingfracture criterion

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to acknowledge the financial support ofthe National Natural Science Foundation of China (Grantsnos 51374228 51274231 and U1262208) and the CNPC KeyLaboratory Fund (Contract no 2014A-4214)

References

[1] B S Aadnoy ldquoInversion technique to determine the in-situstress field from fracturing datardquo Journal of Petroleum Scienceand Engineering vol 4 no 2 pp 127ndash141 1990

[2] M Bloch C A M Siqueira F H Ferreira and J C JConceicao ldquoTechniques for determining in-situ stress directionand magnitudesrdquo in Proceedings of the Latin American andCaribbean Petroleum Engineering Conference SPE 39075-MSRio de Janeiro Brazil August-September 1997

[3] J Huang D V Griffiths and S-W Wong ldquoIn situ stress deter-mination from inversion of hydraulic fracturing datardquo Interna-tional Journal of Rock Mechanics and Mining Sciences vol 48no 3 pp 476ndash481 2011

[4] J Djurhuus and B S Aadnoslashy ldquoIn situ stress state from inversionof fracturing data from oil wells and borehole image logsrdquoJournal of Petroleum Science and Engineering vol 38 no 3-4pp 121ndash130 2003

[5] B S Aadnoy ldquoIn-situ stress directions from borehole fracturetracesrdquo Journal of Petroleum Science and Engineering vol 4 no2 pp 143ndash153 1990

[6] C A Barton D Moos P Peska and M D Zoback ldquoUtilizingwellbore image data to determine the complete stress tensorapplication to permeability anisotropy and wellbore stabilityrdquoThe Log Analyst vol 38 no 6 pp 21ndash33 1997

[7] H Zhao F Ma J Xu and J Guo ldquoIn situ stress field inversionand its application in mining-induced rock mass movementrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 53 pp 120ndash128 2012

[8] Z D Chen Q A Meng and T F Wan ldquoNumerical simulationof tectonic stress field in Gulong depression in Songliao Basinusing elasitic-plastic increment methodrdquo Earth Science Fron-tiers vol 9 no 2 pp 483ndash491 2002 (Chinese)

[9] D K Singha and R Chatterjee ldquoGeomechanical modelingusing finite element method for prediction of in-situ stress inKrishna-Godavari basin Indiardquo International Journal of RockMechanics and Mining Sciences vol 73 no 5 pp 15ndash27 2015

[10] G Maier R Ardito and R Fedele ldquoInverse analysis problemsin structural engineering of concrete damsrdquo ComputationalMechanics vol 9 pp 5ndash10 2004

[11] G Maier M Bocciarelli G Bolzon and R Fedele ldquoInverseanalyses in fracture mechanicsrdquo International Journal of Frac-ture vol 138 no 1ndash4 pp 47ndash73 2006

[12] F Yin and D L Gao ldquoMechanical analysis and design of casingin directional well under in-situ stressesrdquo Journal of Natural GasScience and Engineering vol 20 pp 285ndash291 2014

[13] S Sakurai and S Akutagawa ldquoReport some aspects of backanalysis in geotechnical engineeringrdquo in Proceedings of theISRM International Symposium (EUROCK rsquo93) pp 1133ndash1140Lisboa Portugal June 1993

[14] L Z Xue Engineering Optimum Programming TechnologiesTianjing University Press Tianjing China 1998 (Chinese)

[15] F Li J-A Wang and J C Brigham ldquoInverse calculation of insitu stress in rock mass using the Surrogate-Model AcceleratedRandom Search algorithmrdquo Computers and Geotechnics vol 61pp 24ndash32 2014

[16] X Z Wang W Q Liu Y G Sun and Y J Ma ldquoField testingof fracturing techanolgy in coal formed gas wellrdquoOil Drilling ampProduction Technology vol 23 no 2 pp 58ndash61 2001 (Chinese)

[17] J Teng Y B Yao D M Liu and Y D Cai ldquoEvaluation ofcoal texture distributions in the southern Qinshui basin NorthChina investigation by amultiple geophysical loggingmethodrdquoInternational Journal of Coal Geology vol 140 pp 9ndash22 2015

[18] Z Meng J Zhang and R Wang ldquoIn-situ stress pore pressureand stress-dependent permeability in the Southern QinshuiBasinrdquo International Journal of Rock Mechanics and MiningSciences vol 48 no 1 pp 122ndash131 2011

[19] P Lancaster and K Salkauskas ldquoSurfaces generated by movingleast squaresmethodsrdquoMathematics of Computation vol 37 no155 pp 141ndash158 1981

[20] AHMedley ldquoStorage of natural gas in salt cavernsrdquo in Proceed-ings of the SPE Northern Plains Section Regional Meeting SPE-5427-MS Conference Paper Society of Petroleum EngineersOmaha Neb USA May 1975

[21] G Han M S Bruno K Lao J Young and L Dorfmann ldquoGasstorage and operations in single-bedded salt caverns stabilityanalysesrdquo SPEProduction andOperations vol 22 no 3 pp 368ndash376 2007

[22] P Berest and B Brouard ldquoLong-term behavior of salt cavernsrdquoin Proceedings of the 48th US Rock MechanicsGeomechanicsSymposium vol 14 Minneapolis Minn USA June 2014ARMA-2014-7032

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of