環境調和型地下空間利用のための地下き裂の動特性に関する研究

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(1900999T 岳尊野瀧)

基地管上幽=1割劫嘩

a)彊音i恥Q) Qq=jO肘rJi凶夢i恥両地虻事態

はしがき

我々は現在多くの環境問題を抱えている.その代表が二酸化炭素問題である･二酸化炭

素濃度は年々着実に増加している.二酸化炭素の濃度上昇の原因は,石炭,石油,天然ガ

スといった化石燃料の消費による.大気中の二酸化炭素濃度を上昇させないためには二つ

の方策がある.第一は発生した二酸化炭素を空気中に拡散させないことである･第二は,

二酸化炭素を排出させないことである.前者は,二酸化炭素の海溝隔離あるいは地下固定

等が考えられるが,安全が保証されている技術レベルに達してはいない技術である･一方

後者は,経済成長が続き化石燃料が使い続けられる限り実現は不可能であるが,有力な選

択肢として,二酸化炭素発生量の出来るだけ少ないエネルギー源を求めるという方策があ

る.すなわち水九地熱,太陽光,風力等の自然エネルギーの使用率を高めていくことで

ある.これらは,いずれもクリーンで再生可能なエネルギーであるが,地熱は,我が国に

おける資源量の大きさ並びにエネルギー産出量の安定性という点から,将来を見渡したと

き我が国の有力なエネルギー源として確保しておくべきものである.現状では国内総設備

容量が60万kW弱ではあるが,将来を期待できる技術レベルにある･特に,天然の貯留層

と人工のき裂群とを組み合わせたいわゆる能動型地熱抽出は有力である.一方,上記第二

のもう一つの選択肢に原子力発電がある.この場合は核廃棄物の地下保管という別の環境

問題に対処する技術の確立が必要となる.この二つの技術,すなわち能動的地熱開発と核

廃棄物の地下保管は,卓越した地下き裂による流体の流動という観点では,全く重なり合

うものである.そこでは,き裂の幾何学的並びに物理的性状の把握が必ずしも万全ではな

いという点が,ボトルネックの一つになっている.本研究代表者は,これまでき裂から放

射される弾性波を利用したき裂諸特性の評価法の方法論について検討してきたが,本研究

は,これを実フィールドに適用することを通して確立することを目的とするものである･

研究組織

研究代表者:林 一夫  東北大学 流体科学研究所 教授

研究分担者:斎藤玄敏  弘前大学 理工学部 助教授

研究経費

平成15年度   1, 100千円 平成16年度   1, 000千円 合 計     2, 100千円

研究発表

(1)学会誌等

S. Ito, R. Hayashi, H･Asanuma and H･ Niitsuma, Fracture Characterization by Using

Resonance of Penny-Shaped Fluid･Filled Crack Model in the Higashihachimantai Field,

Geothermal Resources Council Transaction, 27 (2003), 689･693.

S. Ito, K. Hayashi, H. Kaieda and Y. Aoyagi, Fracture Characterization by Using

Resonance of Penny-Shaped Fluid Filled Crack Model in Australian HDR Field - Initial

Results, GeothermalResources Counc止Transactions, 28 (2004), 223･226.

(2)口頭発表 伊藤 伸,林 一夫,浅沼 宏,新妻弘明,東八幡平フィールドにおける流体の圧入よる

岩体の速度変化を考慮した地下き裂評価に関する研究,日本地熱学会平成15年学術講演会

講演要旨集, 2003, 88.

伊藤伸,海江田秀志,青柳恭平,林 一夫,オーストラリア･クーパーベイスンにおける

地下き裂評価に対する物性値の影響に関する研究,日本地熱学会平成1 6年学術講演

会講演要旨集, 2004, A32.

K Hayashi and S･ Ito, A Dynamic BIEM for Cracks Filled with Fluid in characterization of a Reservoir Crack for Advanced Geothermal Heat Extraction, 8th

Us National Congress on ComputationalMechanics, Austin, US, (2005) to be presented

研究成果

き裂の同定において,最も必要とされるものは,き裂の開口変位,き裂の大きさ,き裂

面接触剛性である.き裂開口変位は貯留層き裂の通水特性に直接結びついており,また,

き裂の大きさは,地下流動の広がりに直接結びついている.き裂から放射される弾性波に

は二種の形態が想定される.き裂自身は震源ではなく他の震源からの弾性波に励起される

ことで発する弾性波,並びに,き裂周りの状況の変化に応じて開口･すべりが発生しその

結果生じるものの2つである.以下,前者を使う同定をアクティブ,後者を使う同定をパ

ッシブという.き裂同定は,基本的には,フィールドで観測された弾性波のフーリエ像空

間のピーク周波数と,理論的に予測される地下き裂の固有振動モードの周波数との比較し

て行う.これまで研究代表者らは,内部を流体で満たされたき裂の動特性に関する理論的

研究を実施し流体の粘性や圧縮率,岩体の弾性係数等の効果に関し定量的に明らかにして

きた. 2年継続の初年度では,まず,岩体内き裂の周囲に損傷域が存在する場合を想定し

て,岩体実体波の位相速度が小さい場合の検討を行って,人工き裂周りの損傷域の影響を

調べた.次に,円盤状き裂の軸対称変形モデルでは表現できない非軸対称変形(例えば,

円板の一つの直径を節とするような振動)をも考慮した定式化を実施し,その数値計算法

についての検討を行った.最終年度では､これまで本研究代表者らが構築してきた方法を

実フィールドに適用して,方法の実用性のデモンストレーションを行った.まず､東北大

学東八幡平実験フィールドに適用した.同フィールドに作成された地下流体循環システム

は2本の坑井(EE4､ Fl)とそれに連結した人工き裂からなっている.計測は以下の

ように行った.近傍に新坑井EE5を掘削する. EE5で発生するドリリングノイズによ

り励起されるき裂内圧の変動を､ EE4とき裂の交点に設置したハイドロフォンによりモ

ニターする､というものである(アクティブ).この圧力変動のピーク周波数と､円板状

き裂モデルの軸対称モードの固有振動数が互いに等しいという条件からき裂特性を求め

た.その結果､き裂の半径は約4 0mである.き裂の初期開口幅は､坑口基準の貯留層圧

2MPaを境にしてステップ状に変化する.貯留層圧が2MPaより小さい場合は約0.3mm､

大きい場合は約0.5mmという評価が得られた.このことは､このき裂が,地表基準で約

2MPaで開口することを示している.引き続き､本方法をオーストラリアのHDRフィー

ルド(クーパーベイスン)で観測された長周期地震に適用した(パッシブ).この地震の

震源深度は約4000mと推定されている.約100秒の長い継続時間を持つ長周期地震であ

ることから､これは､内部を水で満たされたき裂に生じた定在波を根元とするものである

と考えて合理的である.解析の結果半径が約160m､初期き裂開口幅が約17mmという評

価結果が得られた.また､き裂面接触剛性は極めて小さく評価された.このことはき裂が

ほぼ完全に開いていることを示唆するものである.

Ⅳ

Part I:

Technological Background and Formulation

ー1-Technological Background

ln advanced geothermalheat extraction, hydraulic stimulation/fracturing is performed in

any cases to activate a reservoir system consisting of preexisting fractures and those induced by hydraulic stimulation/fracturing. Such a reservoir system is called EGS (Enhanced/Engineered

Geothermal System)･ EGS includes HDR (llot Dry Rock) geothermal heat extraction. The world

first HDR project started in the early 1970's byLeSAlamos National I.aboratories in Fenton Hill,

NM under the sponsorship ofDOE (Duchene, 1991). HDR is the incentive for development of the concept of EGS and for this reason EGS is sometimes called HWR (Hot Wet Rock) Takahashi and

Hashida, 1992). HDRgives us a very sharp geometrical and physical image, leading us to clear

understanding of the basic mechanism of geothermal heat extraction, So that we use HDR model

for discusslng the topics concemed. In a HDR system, we have to create afluid circulation

subsurface system, which consists of two wells and a reservoir created/stimulated hydraulically.

One of the two wells is the injection well and the other is the production well. Water is pumped

down throughthe injection well,flows into the reservoir, extracts heat from the surrounding rock

by directly contacting with the rock through fracture surface and goes up to the surface throughthe

production well (Hayashi and Abe, 1987). Obviously, whether the wells are connected to the

reservoir or not is our prlmary COnCem. Therefore, it can be said that one of the key technologleS is the characterization of the reservoir血actures created by hydraulic fractming/stimulation. It is

critical to estimate the basic geometrical characteristics such as location and extent of reservoir, size of fractures, aperture of fractures and degree of contact between the two faces of each fracture･

All these are closely related to localhydraulic properties of reservoirs･ For this purpose, seismic

methods have been considered to be most promlSlng. Seismic methods provide only mean of

obtaining reliable information on the interior of the reservoir･ It should be noted that the methods

are sometimes called AE/MS (Acoustic Emission/Micro Seismic) methods･ The seismic methods

have been verified to be powerful in the characterization (Niitsuma, et a1., 1987;1999) and have

been widely used (e.g., Fehler and Bane, 1985).

Here in this report, We divide the methods into two groups, passive and active･ The term

passive seismic is taken from earthquake seismology and refers to methods in which seismometers are used to monitor earthquakes･ Passive seismic monitoring has been used for in-situ monitoring of hydraulic fracturing/stimulation in almost every EGS field. It isthe principal mean by which

-2-the creation and growth of hydraulically stimulated geo-2-thermal reservoir can be monitored in real time･ Fault plane solutions, Comer frequencies and moment tensors are the examples of parameters

for the characterization of seismic source. Methods have been proposed to investigate whether

structures can be identified within the cloud of micro seismic location data. Two of these methods,

known as the three-point (Fehler et a1., 1987) and collapsing method, (Jones and Stewart, 1947) are

statistical techniques. The other methods, such asthe doubletanalysis (Poupine et all, 1982) and

cluster analysis (Phillips et a1., 1997), improve the reliability of relative locations･ The results of

applying these techniques show that some predominant structures existina man-made reservoir･ If naturalor induced seismicity is observed inthe vicinity of a potential rock mass, it is also used to

characterize the reservoir. Since a passive seismic slgnal is the by-product of subsurface phenomena and does not directly represent a physical property of the reservoir, an interpretation of the meanlng Of seismic activity is essential･

Active methods have been developed as complements to the passive methods discussed

above. The methods employedinclude cross-well seismic, vertical seismic pro filing (VSP), tube

wave analysis, crack wave analysis and measurement while drilling, all of which provide

information on mechanical properties, location and presence offractures in EGS reservoirs･ The

simplest way of active methods is to detect response of a reservoir crack activated by elastic wave from anartificial seismic source, such as electric sparkers, explosives and air guns, as well as drilling noise induced at bottom hole of a near by well under drilling･ If it is possible to set a

detector at an intersection point between the well and the reservoir, the active method is useful

because the pressure change in the fracture induced by the artificial seismic wave can be observed directly under very controlled conditions･

In order to estimate the characteristics, We need to understand the dynamic response of a

fluid-filled crack･ The dynamic response of afluid-filled crack has received intensive attention for

understanding the source mechanism of volcanic tremors and microseismic events observed during hydraulic fracturing (C.g., Chouet, 1986)･ In these studies, reservoir crack was treated as an

infinite fluid-filled slit or a single crack surrounded by a rock･ However, in EGS fields, geothermal

reservoirs consist of networks of cracks and permeable layers, suggesting that fluid leakage from

the crack periphery is likely to be one of the main factors govemlng the dynamics of a批id-filled

crack in EGS.

-3-Fracture characterization of the reservoir cracks was camied out by uslng many Crack

models･ For example, Ferrazzini et al. (1990) examined long-period events, which were observed during hydraulicfracturing at the Fenton Hill HDRfield･ They employed a rectangular planer

crack model proposed by Chouet (1986) and estimated length, width and thickness of reservoir

cracks at the Fenton Hill HDR field･ Nagano and his coworkers observed guided waves in the

reservoir crack created by hydraulic fracturing in the Higashi-Hachimantaifield, where the

reservoir crack was connected to two wellbores (Nagano et al., 1995; Nagano and Niitsuma, 1996). They set an alr gun at One Of the intersection points between the crack and the wellbores and set a

triaxial seismic detector at the other point of intersection, and furthermore, they estimated the interracial stifhess due to contact between the asperities on the upper and lower surfaces of the

crack by using the model of Hayashi and Sato (1992).

-4-Formul atio n

Afluid-filled crack is an appropriate model for examlnlng the behavior of subsurface

fractures, and has attracted much attention, such as growth of geothermal reservoir, mlgration of

water in aglacier and transportation of magma in the earth's crust･ Regarding the dynamics of a

fhid-filled crack, however, ourknowledge is rather limited to 2D･ Even if the fluid-filled crack is

of penny shape, literatures are available only for pseudo 3D cases that the deformation is axISymmetric withrespect to the centerline of the crack･ Suppose a penny shaped crack is

oscillating. If we allow onlythe axISymmetric defomation, then the nodes of the fundamental

modes ofoscillation of the crack surfaces are likely to be as shown in Figure I-1･ On the contrast to

this, if we allow real 3D deformation, there must be fundamental modes shown Figure I-2 that is

completely different from the axISymmetric case･ This does not necessarily mean that the characterization utilizing only axISymmetric modes leads to fatalmisunderstanding of in-situ

phenomena in a rock at a depth･ So far, we do not have any quantitativeknowledge on this issue･

Under these considerations, we construct, here in this chapter, the basis for 3D approach based on

BIEM in Laplace and Fomier image spaces･

Letusconsider a dynamic problem ofa planer crack in an infinite elastic body･ Introduce

a cartesian coordinate system o -xl,X2,X3 With the ongln Onthe crack and thex3-axis being normal

to the crack･ The displacement and the stress are denoted as ukand qiJ･ , reSpeCtively･ Here and in

thefollowlng the Latin indices take on value 1, 2, 3, and the usual summation convention on

repeated indices are employed, unless otherwise stated･ The crack is under tensile mode

deformation･ Then, the displacement gaps across the crack plane Au.･L= u.l.,ーd -u.･｣｡) are zero

except for Au, ･ The boundary condition on the crack surface is

cr,,｣｡ ≡ 03'3

(for (xk )∈A')      (Ⅰ-1)

where a,',(xl,X2,i) is normal stress applied on the crack surface A'･Let盲(p) be the IAplace transformofafunction g(i) of time i as follows:

i(p) -I."g(tPdE ･

consider an infinite elastic body without the crack subjected to an impulsive unit load

applied to a point Q (X) at 1 - 0 in the x)I direction･ The displacement in thexE direction at P( e ) is

glVen by

6Ltj'(X･f,p,-*efa･･,･-fIie苦-iefIjJl (I-3,

where ( ),I ≡∂( )/aEL ,R-tx一句and CL and C'arethe phase velocities of P and S waves, respectively･ The phase velocities are glVen by

1 1

cL -(if)ち, cT -(;)i     (I-4'

where ∂,･J is the Kronecker delta and p is the density of the elasticbody. The quantities入and FL

are the Lame constants.

Let us apply the reciprocal theorem in the I･aplace image space (see appendix) between

the solution of the problem just discussed above and that of the original crack problem･ Then we

have thefollowlng expression of the displacement of the crack problem:

dJ･ (X,p) - -JA. Ad3(i,P)･軒3j'(X, i,p)rdA(i)    (I-5)

f'3)｡ - -C33〟孟甜         (I-6'

where

cEju - A∂.･j∂〟 ･ FL(aJ･k∂.･, ･ ∂)･,∂止)       (I-7)

Here, ( ･ )'denotes the value as x,-+0. Hooke's law and displacementgiven above lead us to

･LJ･ (X, P) - -I^･ Au-3(i,p).C･･)･u新i"(X, i,p)YdA(i)  (I-8,

Therefわre the Stress On the crack surface are glVen by

6L･j(X･PL.0 --pfIA･Au-3(f･p,･C･･).uliEy'(X･f･p')'｣｡-'f'(X∈A･) (I-9,

where Pfl denotes the finite pad integral･ Finally we amive at the following expression for the

normal stress on the crack surface:

-6-言33(X･P1.,→.0 ≡ -pfIA. Au-擁66111'1 ･瑚･ (A ･ 2p鵜

.磨.,U-.'27!2).(A.2握,) ､.

- リJ=ぷ

. (At. 2廉さ≡

〟_＼ ー

.16'23!,-)｣(A.読望乱.｡dA (X∈A･)

The above expression rewritten as fわllows:

633(X･P1.,-.0 - -ipfIA･ Au3(fw4%(E)2 fe-pF/CT - 4(叫(1 ･ E)e-pr-/C,

-i lA2 it(a)4e-pF/C, -(a)4e-pF/CL I

-吉軒- -(吉日

- 4p(叫((a)2e-p7/CT - (a)2e-pF/CL i

･.2p2 i((吉)e-pr-/C, - (吉)e-pF/CL I ･12p2g(e-pF/CT-e-PF/CL)])dA(i) (X∈A･) where r∼- (X1-51)2-(X21E2)2

Let us introduce thefollowlng nOn-dimensionalquantities:

=-, X=-, y=生, i-を,r Xl

a a a a

4--坐

α

Then Equation (Ⅰ-1) becomes

諺, S=買, k-ら

a Cp -7-(I- 10) (I-ll) (I- 12) (I-13)

ま,,(X,y, S) 一志pfIA. i(如･ S)･

((fl2)2f-4(嘉一1)(;･叶r

-ii(i-2)2f(e-S, -k4e-hr)

-4(i-2);(e-sr -k3e-hr)

-4(嘉一3)i(e-sr -k2e-br )

･12;(e-S'-ke-br )

･12吉(e-sr-e-也,))]dA(E･q) (X･y∈A･) (I-14)

The integrand is slngular, so that we divided it into two parts, i･e･ slngular and non-slngular parts･ Then, we have

633(X,y,S)

_{-ipfIA･i(i,∼,中吉･芸･子)a(5,q)}

- fIA･i(5,∼,S,lIl 4(2 ･ sr,i -1259)e-sr ･

9 1 65∫2 +-●---●-2 r3 32 r ]u(i,∼) (X,y∈A･) 1+l∫′

蛋-T･12-チ

1 --∫r e2 (Ⅰ-15) In deriving Equation (I-15), we have set the Poisson's ratio( V ) to be 1/3. The ratiok(- C,/CL ) is

1/2.

It is readily understood that the problem is reduced to solving the singular integral

equation deduced from Equations (Ill) and (I-15), unknown function of which is the displacement

gap across the crack ￠ ･ It is impossible to get a closed formsolution of the integral equation. In

the followlng, We specify the shape of the planet crackand discuss the way to get numerical

solutions, which is specific to the shape of the planer crack.

ー8-Let the shape of the planer crack be elliptic. The crack is filled with fluid. Its major and

minor axes are 2a and 2b, respectively･ The Cartesian coordinate system o -xl,X2,X, lS Centered at

the elliptic crack. (FigureI-3). With the aid of the asymptotic behavior of the displacement gap in

the vicinity of the crack periphery, let us set such that

i(X,y,S)=

1-X2 -篭慧AmnTm(xhn(三)   (Il16,

where Ann is unknown constant to be determined. The function Tm (m=0,1,2 ･･･) are the

Chebyshev polynomials of the first kind. The parameter c is the aspect ratio defined by b/a･ By

substituting Equation (I-16) into the singular integral equation, we obtain afunctional relationship

that must be satisfied on the crack surface. We apply a usualCollocation technique to determine

the unknown constants sothat the functional relationship is satisfied at a set of points, that is

selected appropriately (Figure I-4). Total number of the collocation points should be J (=MXN) since the total number of the unknown constants isJ. It should be notedthat the fhite part integral

included in the functional relationship are evaluated by

pfIA･ :dA - -f蒜        (I-17'

pflA. fdA -f α(函         (I-18)

where the function a(a) is the distance from a point on the crack to the periphery (Figure I-4) and

isglVenby α(♂)≡ cos2β sin2β + β    β

苧･苧

苧･と型

b2

)21(S･笥(妄･t-1)i

(I-19)

By applying the collocation technique, we arrive at simultaneous linear algebraic equations that

can be expressed schematically as fわllows:

b,･ - a,.lEl ･a,･2E2 ････a.I)･E,I ････au_1EJ_1 ･auEJ (i =1,2,･･･,J)  (I-20)

bE = i,,(xL, y"P)/IL      (I-21)

Ej ≡Amn ()'-Nm･n･1)      (I-22)

a.i ≡IA. Tm (紘(∼)g(xE･ y〟的･S極q         (I-23)

-9-where g(xE, y.･, 5,77, S) is aknownfunction. The simultaneous equation also express in a form such

that

all aJ2

The solution of which is glVen by

E,-諺(j-L2,3,.･･J)

Cノ(∫)-aJ1 ... aJJ-1

Thus, the solution is glVen by

i(X,y,S)-

1__t･, _ど

C2

童謡霊芝Tm(xh(三)

In the time domain, crack opening displacement is expressed as

Q(X･ y･ I ) - aL'.1':聾esTds

where I(X,y,S)-

1-X2 -g真cj(S)霊芝Tm(X>n(f)

(Ⅰ -24) (I-25) (I-27) (I-28) (Ⅰ-29) (I-3 1)

In Equation (ll29), T is non-dimensional time defined by CTt/a and h is a real number larger than

the real part of every zero point of A(S).Letsk (k - 1,2･･･) be the zero point. Then it follows that

A(S)-白(S-sk) -10-(I-32) ら , Q r ` ･ ･ ･ れ 一 _ 2   2 2 ︰ ● β     α l 1 I       ▲ ソ 一   ● ● ● 〃     α I J   打 . . .   〃 α     d α L i ) . L i l ( . ･ ･ ･ ち l 一           1 l       一 ･ 1         . I 1         2 8     d l         1 1         2 α α U   〃 ︰ .   〃 α     α           8 7   7         + ノ   . ノ ︰ ･   〃 l         つ 一 α     α         α ら れ 一 ･ ･ ･ 九 .

With the aid of the residue theorem, we get

∫

Q(X･ y･T) -芸Res(sk ksk' (I-33)

It is readily understood that the real and imaginary Parts Of sk give the intensity of attenuation

and the eigen angular frequency the k-thfundamental mode, respectively･

-il円-Figure I-1･ Schematic presentation of axisymmetricfundamental mode of

oscillation. The solid and broken lines represent the crack periphery and nodes of oscillation, respectively･

Figure I-2･ Same as Figure I-1for non-axISymmetriCfundamental modes of

oscil lation.

-12-Figuret･3･Ellipticcrackandcoordinatesystem･

-13-i,一一-｢一一Jー ● / ＼＼ I/ノr l  1-ヽ/ _〉( ･ l ' /X＼ I′′ '＼〉′--ll-㌔-Y/ ヽ -J ′ _ ! . ＼､ _ ∫/ ●/′ ■′一一一一一 ● l   /   ′        ＼    ヽ   l   / ∫ ′ ＼ I l

十 十 (､ ● 十十十

＼ _､､ ＼＼∼___ノ/′ ノ ′

′一ヾ ､ゝ'/｢ン// ●′L

ヽ､ ,/ ､一一J__一一,､＼ /

′二上､･ト:こく

l ′ト一十一一で

Figure I-4･ Example of selecting collocation points･

q (Ⅹ,y) ; 鋳

Ir α

Figure I-5 GeometriCalpresentation of a ( 0 ).

-14-l t

Part II:

Application to Fields

ー15-Axisymmetric Formulation

Penny-Shaped Fluid -Filled Crackunder AxisymmetriC Deformation

We neglect the effect of fhid leakage along crack periphery･Let us consider a

penny-shaped hid一色lled crack of radius a and of aperture d in an infinite elastic medium, i･e･ rock (FigureIIl1). The elastic medium is under axisymmetric defomation with respect to the centerline

of the crack.Let us introduce a Cartesian coordinate system (xi) (i=1, 2, 3) and a cylindrical

coordinate system (r, 0, X3). The stress and the displacement, referred to the coordinate system are

denoted as q,･). and ui (i, j=1, 2, 3), respectively･ In thefollowing, the stifhess permit area induced

by the partial contact between the upperand lower crack surfaces is denoted as k･ We call the

stifhess "interfacial stiffness''.Let g-b) bethe Laplace transform of afunction g (i) of time t･ The

boundary condition on the crack surface is expressed asfollows:

示,,l勺_A-kAd,一戸･ii,', (r ･a),         (ⅠⅠ-1)

where Au3 is the gap of u3 induced along the crack plane and P is the pressure of the fluid in the

crack. The stress o;, is the nomalstress applied to the crack surface or introduced to express the

crack growthfromthe crack of radius a'to that of radius a･ Expressions of o33 and P are not

presented here for brevity (see Appendix)･

Let us introducethefollowlng nOn-dimensional quantities:

R-工, S=貰, 80-坐,亭-ら生

a LA a2

(IIl2)

where l▲ is the shear modulus of rock and CT is the phase velocities of the S-waves of the rock･ From Equation (IIll), We get a singular integral equation of the non-dimensional displacement gap

after tedious manlPulations･ It is impossible to solve the singular integral equation analytically･

Here, taking account of the asymptotic behavior of the displacement gap along the crack periphery,

we express the non-dimensional displacement gap asfollows, with unknown coefficients An and

the Chebyshev polynomial of the second kind Un(･):

6(R,S) - JiT蓋A-A (S)U2k-. (R)･     (II-3'

By substituting Equation (II-3) into the singular integral equation and by applying a usual

collocation technique, we fhally amive at the simultaneous linear algebraic equations for A" (n=1,

一16-2, ･.･, N). We introduce a function A(S), which is determinant of the matrix, the elements of which

are the coefficients of the simultaneous linear algebraic equations･ The imaglnary Parts Of the roots of A(S)=O are the eigen angular frequencies･ The real parts of the roots of A(S)=O represent the

intensity of the attenuation of the oscillation. In thefollowing, we denote the roots of A(S)=O as cLj

and the imaglnary Parts Of cLj aS E2)･･ Here,J represents the mode number･ It should be noted that the

complex conjugate of q･ is also the root of A(S)=O and s=cLj is the zero point of order l･

Following Hayashi et al. (1995), we define the non-dimensional interfacial stiffness ao such that

80 - 8乙叢,        (I1-4'

where

aL - 2(.･V)告         (II-5,

Here, v is the Poisson･s ratio of elastic medium,A'isthe realContact area on the crack andAo is the

area of the cracked reglOn.

Characterization

ln order to estimate crack radius a, aperture d and interfacial stiffness aも, we use elgen

frequencies derived from the crack model and the peak frequencies obtained from field data･Let us

denote peak frequency ofthej-th mode determined from field data asfeJ･･ In view of Equation (II-2),

wehave

fcj売手      (.I-6)

wherefcj is the eigen frequency ofj-th mode calculated･ Obviously E2j is afunction ofa, d and a乙･

Thus, ifa, dand aとare set to be true values in calculating E2j, We Should havefcJ･ =fej･

For characterization, We need to estimate mode numbers of peak frequencies obtained from field data･ Now, let us discuss briefly how to estimate the mode number of each peak frequency observed in field data･ Detailed inspection of E2j Calculated by uslng the present model leads us to the fbllowlng relationships:

E2)I.2-E2j.1≧E2)I.1-E2j (j≧2)

E2, / E22 i C, ,       (ⅠⅠ-8)

where C, is equal to 1.8 in the case of the present model･ Non-dimensional frequency E2ej (defined as 23'afei/CT) should satisfy the conditions which are obtained by replacing E2j by E2ej in Equations

(II-7) and (II-8). Therefore, we have to select peaks from the field data in the frequency domain

and allocate mode numbers to selected peaks so as to satisfythe two conditions just stated･ lf such

a selection is impossible, We should quit to apply the present method to the field data･

Based upon these considerations, we introducethefollowl･.g Objectivefunction Er and reduce the problem to finding a set of parameters a, d and aもby optimizing Er:

Er -蓋(fe･･ - fcE(a･d･aJ)2       (ⅠⅠ-9,

Here, J and N are the smallest mode number of peak frequency and the largest mode number found from the field data. Marquardt method is employed to minimize the objective function Er, we find

a set of parameters.

Application to Field I- 1･ IIigashi･HachimantaiField書

FigureII-2 shows the schematic representation of set up･ In this field･ a subsurface crack

was created by hydraulic fractming at a depthof 369m in well F-1 in 1986･ After hydraulic

血acturing, well EE-4 was drilled so as to intersect the artificial subsurface crack at a depth of 360m･

Distance between the intersection points of wells F-1 and EE-4 is about 7m･ This artificial

subsurface crack is createdinhighly welded tuff. In this report, We characterize the subsurface

crack byuslng the data of crack waves observed by a hydrophone during drilling a nearby well

(well EE15) in 1993. The subsurface crack was stimulated by drillingmise (elastic waves) from

the bottom of well EE-5. The subsurface crack was pressurized to change the interfacial stifhess

of the crack. Wellhead pressure was monitored in well F-1･ Niitsuma (1989) reported that the

wellhead pressure of F-1 wasalmost equalto that of well EE-4, so that we chose the well head

pressure of well F-1 as a reference of the reservolr PreSSure･

In order to solve the inverse problem to estimate crack size and so on,followlng Values

'This is excerpted from lto et al. (2002q, 2002b, 2003)･

-18-have been used, referring to material properties of water andthe welded tuff: the ratio of the rock density to fluid density is 2･6,the ratio of shear modulus of the rock to the bulk modulus of water is

3.6, the hid viscosity yl is 10 3 pa･S (1 cP), the porosity of the rock is 0.01 and permeability of the

rock is 10-19m2･Also, the phase velocities of the P- and SIWaVeS Of the rock are 3000m/s and

1750m/s respectively (Nagano and Niitsuma, 1996)･

Figure ll-3 shows example of amplitude spectra of the field data observed during drilling

of the well EEl5. Pw is the wellhead pressure of the well F-1. The peak frequencies (circled) are

used for characterization･Anows show calculated peak frequencies derived from the crack model

by uslng estimated parameters･ By uslng thepeak frequencies of each wellhead pressure, we can

estimate crack size, aperture and interfacialstifEness･ Results of characterization are shown in FiguresII14, 5 and 6･ In these figures, the results of the case that the wellhead pressure is zero are also presented･ Figure Il14 shows estimated radius of the crack with respect to the wellhead

pressure･ In the case that the wellhead pressure is less than 2MPa, estimated crack radius is around

40m･ For the wellhead pressure larger than 2MPa, estimated crack radius is around 20m or less･

Figure lI-5 shows variation of estimated aperture･ In the case that wellhead pressure is less than

2MPa, estimated crack aperture is around O･3mm･ For the wellhead pressure larger than2MPa,

estimated crack aperture is around O･5mm･ Figure Il-6 Shows variation of estimated interracial

stiffness 8乙. In the case that wellhead pressure is less than 2MPa, estimated interfacial stiffness is

around O･l or above･ For the wellhead pressure larger than 2MPa, estimated interfacial stiffness is

less than 0.03. 1t can be seen from these figures thatthe geometrical/mechanicalCondition of subsurface crack changes stepwise at a value of the wellhead pressure between l･5MPa to 2MPa･

Radius of the subsurface crack is estimated to be around 60m. This estimate is derived from the muid volume injected into the crack during hydraulicfracturing (Niitsuma, 1989)･ Radius

derived from transmissibility test is about 35m (Hayashi and Abe, 1989)･ In the case that the

wellhead pressure is lower than2MPa, present results of radius agree with the results just

mentioned above. But, in the case that the wellhead pressure is higher than 2MPa, the crack radius

obtained here is smaller than the results just mentioned above. From transmissivity test (Hayashi and Abe, 1989), the transmissivity of the crack changes with reservoir pressure smoothly･ TYLis means crack aperture changes with wellhead pressure smoothly･ But, the present results do not show similar tendency as that of transmissivity･ The estimated aperture changes stepwise with the

ー19-wellhead pressure･ The magnitude of aperture estimated here is largerthan the results obtained by Hayashi and Abe (1989). Saito and Hayashi (2002) estimated the aperture by using Stoneley wave･

Magnitude of aperture estimated from Stoneley wave is twice as large as the present results･

Interfacial stifhess is small when aperture is large, because contact area between upper and lower

crack surface becomes smaller with crack opeming (Equation (II-5)). Naganoand Niitsuma (1996)

estimated interfacial stiffness by setting crack wave (guided wave) velocities to be O･001 or less･ They observed crack wave velocity between the two intersection points (see Figurel1-2)･ This

means that interfacial stifhess estimated by Nagano and Niitsuma (1996) is on crack wave path･

Present results of interfacial stifhess is larger than O･001･ It is understood that the present results

show some kind of average of interfacial stifhess in the area of the crack, which includes crack

wave path.

From the results discussed above, it can be said that the present characterization method can estimate factors inanarea in which resonance of the crack wave takes place･ In the case of low wellhead pressure, radius of area of resonance is around 40m and the true crack radius is larger

than present results. In the case of highwellhead pressure, radius of area of resonance is around

20m or less･ It is not clear how the area of resonance changes and why the condition of the crack

changes stepwise with respect to the wellhead pressure･

The results obtained from the study of lligashi-Hachimantai field discussed above are

summarized as fわllows:

(1) The estimated crack radius is around 40m･

(2) In the casethat the wellhead pressure is less than 2MPa, the estimated crack aperture is around

O･3mm･ For the wellhead pressure larger than 2MPa, the estimated crack aperture is around

O.5mm in the area of radius around 20m.

(3) In the case that the wellhead pressure is less than 2MPa, the estimated interfacial stiffness is around O･1 or above･ For the wellhead pressure larger than 2MPa, the estimated interracial

stifhess is less than 0.03 in the area of radius around 20m.

(4) In this field, the estimated factors (e.g., crack aperture) change step wisely with respect to the

wellhead pressure at around 1.5-2MPa.

(5) This characterization method can estimate parameters in the area of resonance of crack wave･

-20-Application to Field I- 2･Anstralian HDR site'

In order to solve theinverse problem to estimate crack size and so on, the followlng values have been used, referring to material properties of water and granite: the ratio of the rock density to fluid density is 2･7) the ratio of shear modulus of the rock to the bulk modulus of water is

10.2, the fluid viscosity Yl is 10-3 pa･S (1 cP), the porosity of the rock is 0.01 and permeability of the

rock is 6xl0-17m2･Also, the phase velocities of the P and S waves of the rock are 5029 m/s and

2903 m/s respectively.

Figure IIl7 shows the locations of AE observation wells in Australian HDR field･ This

field is located in the Cooper Basin) South Australia･ Geodynamics Limited is carrylng Out the Hot

Dry Rock project in this field. CentralResearch Institute of Electric Power Industry, Japan, has

participated to observe acoustic emission by uslng the Wlde Area AE Monitoring System･ Wells of

this system are WAl, WA2, WA3 and WA4inFigure II-7･ The natural frequency of seismographs

in these wells is around 5 Hzand waves are digitized at 1 KHz (Kaieda et all, 2003)･ We use data

observed at wells WA2, WA3 and WA4.

Figure II-8 shows full wave forms of the event observed at well WA4, at around 23:30 on March 14, 2003, during the drilling of a well in this field･ We applied a low-pass mter with a cutoff

frequency of 250 Hz･ This event is a weak attenuation event and continued over 100 seconds･

Epicenter of this event is shown in Figure II-7 and depth of hypocenter is roughly 4000m･ The

location error is roughly estimated about 100 m. The large location error is due to the situation that

onlyfour stations were operated at that time･ In addition, no S wave phase was detected in the

waveform･ Four of these long period events, including the event shown in Figure II-8, were observed in one observation week. After nine months, the measurement system had been operated for two months. But no such events were observed. Figure II-9 shows amplitude spectra of the field data observed at wells WA2 and WA4. Peak frequencies (circled) are used for

characterization. Peaks between 3 and 6 Hz and > 7 Hz are not chosen because spectra in these

ranges reflect the effect of natural frequency of seismographs and background･Anows show

'This is excerpted from Ito et al. (2004).

-21-calculated peak frequencies obtained from the present crack model･

lt was found that there was a penny shaped crack of radius about 160m with aperture of about 17mm. Theinterfacialstifhess a乙was estimated to be about 6.5Ⅹ1014. This value of interfacial stiffness means that the ratio of contact area to crack area is very small (Equation (II-5))

and that the crack opensalmost completely･ Upon solvingthe inverse problem, We found some

several sets of parameters a, d and ∂ 'Could make E, zero on the same level of accuracy, 1mPlying

that solution is not unique. Thus the objective function E, was not appropriate enoughtogive a

unlque SO)ution･ For this reason, We picked up the intensity of attenuation･ By uslng Our Crack

model, we can calculate the intensity of attenuation of each mode･ We can also calculate intensity of attenuation of waveformS, such asthose showninFigure Il-8, and compared it withthat derived

from the present crack model･ Then, we might have a deterministic solution･ But the former is far

larger then the latter in the present case･ Thus, the inversion results are not so reliable･ We must

introduce additional conditions to constraint the solution.

The results obtained from the study of Australian HDR field discussed above are

summarized asfoIlows:

(1) The estimated crack radius is around 160m.

(2) The estimated crack aperture is around 17mm.

(3) The estimated interracial stifhess is very small indicating thatthe ratio of contact area to crack

alea is very Small.

-22-Acknowl edgments

The field data of Higashi-Hachimantaiwere supplied by Drs･ Asanuma and Niitsuma, Tohoku University･ The field data of Australia HDR site were observed by the collaboration work of the Central Research Institute of the Electric Power Industry, Japan, and Geodynamics Limited, Australia.

Referen ces

Chouet, B･, 1986･ uDynamics of a Fluid-Driven Crack in Three Dimensions by the Finite

Difference Method", ). Geophys. Res., 91, 13967-13992.

Duchene, D･, 1991･ ulntemational Programs in Hot Dry Rock Technology Developmentn, Geother･

Resour. Coun. Bull., 20, 135-142.

Erdogan, F･, 1978･ HMixed Boundary Value Problem in Mechanicsn, ed･, Nemat-Nasser, S･･

Pergamon Press, 4, 1-86.

Fehler, M. and Bane, D., 1985. "Characteristics of Microearthquakes Accompanying Hydraulic Fracturing as Determined from Studies of Spectra of Seismic Waveformsn, GeoEher･ Resour･

Coun. Trams., 9-II, ll-16.

Fehler, M. C., House, L., Kaieda, H., 1987. Determining Plane alongWhich Earthquakes Occur:

Method and Application to Earthquakes Accompanying Hydraulic Fracturing･ J･ Geophys･

Re5., 92, 9407-9414.

Ferrazzini, V. Chouet, B., Fehler, M. and Aki, K., 1990. "QuantitativeAnalysis ofLeng-Period

Events Recorded during Hydro血acture Experiments at Fenton Hill, New MexicoM, J･

Geophys. Res., 95, 21871-21884.

Hayashi, K･and Abe, H･, 1989･ uEvaluation of Hydraulic Properties of the Artificial Subsurface System in Higashi-hachimantai Geothermal Model Field", Journal of the Geothermal

ー23-Research SocietyofJapan, ll, 203-215.

Hayashi, K., Niiyama, K. and Abe, H., 1995. "Dynamic Response of a Rock Mass with a urge

Fluid-Filled Crack", Mechanics of Jointed and FaulEed Rock ed･, Rossmanith, H･ R,

Balkema, 699-705.

Hayashi, K. and Sato, K., 1992. "A Theoretical Study of AE Travelhg througha Fluid-Filled

Crack with Application to Characterization of a Geothermal Reservoir Crack'', Progress in

Acoustic Emission lq, ed., Kisht., T et al., Jpn. Soc. for NDI, 423-430.

Ilo, S･, Hayashi, K･,Asanuma, H･ and Niitsuma, H･, 2003･ uFracture Characterization by Using

Resonance of Penny-Shaped Fluid-Filled Crack Model in the Higashi-Hachinantai Field,

Japan", Geother. Resour. Coup. Trams. , 27, 689-693.

Ilo, S･, Hayashi, K･ and Saito, H･, 2002a･ uElastic Waves Radiating from a Penny-Shaped

Reservoir Crack", Geother. Resour. Coun. Trams. , 26, 303-308.

Ito, S･, Hayashi, K･ and Saito, H･,Asanuma, H･ and Niitsuma, H･, 2002b) uApplication of

Dynamic Response of a Penny-Shaped Fluid-Filled Subsurface Crack to Fracture

Characterization in Higashi-HachimantaiField", 7Transactions of the Japan Societyof Mechanical EngifWerS, 68A, 1460-1465 (in Japanese)･

Jones, R･, H･, Stewart, R･, 1997･ A Method for Determining Significant Structures in a Cloud of

Earthquakes. J Geophys. Res. 102, 8245-8254.

Kaieda, H, Aoyagi, Y･ and Wybom D･, 2003 HWide Area AE Monitoring System in Australian HDR Project", 25th anniversary Natural Energy Forum and 2003 Annual Meeting

Geothermal Research SocietyofJapan Abstracts with Programs, 79 (in Japanese)･

Niitsuma, H., 1989･ uFracture Mechanics Designand Development of HDR Reservoirs - Concept and Results of the T-Project, Tohoku University, Japan", LnE. J･ Rock Mech･ Min･ Sci･ &

Geomech. Abstr., 26, 169-175.

-24-Niitsuma, H., Chubachi, N. and Takanohashi, M., 1987. "Acoustic EmissionAnalysis of

Geothermal Reservoir and its Application to Reservoir Control", Geothermics, 16, 47160･

Niitsuma, H., Fehler, M., Jones, R., Wilson, S.,Albright, J., Green, A., Baria, R., Hayashi, K.,

Kaieda, H., Tezuka, K., Jupe, A., Wallroth, T., Comet, F･, Asanuma, H･, Moriya, H･, Nagano,

K･, Phillips, W･, Rutledge, J･, House, L･, Beauce, A･,Aide, D･, and Aster, R･, 1999･ uCurrent

Status of Seismic and Borehole Measurementsfor HDR/HWR Development", Geothermics,

28, 475-490.

Nagano K･ and Niitsuma H･, 1996･ 〟Crack Sti飽IeSS丘om Crack Wave Velocities'', Geophysical Research Letters, 23, 6891692.

Nagano, K., Saito, H. and Niitsuma, H., 1995. "Guided Waves Trapped in anAnificial Subsurface

Fracture", GeoEher. Sci. and Tech., 5, 63-70.

Phillips, W･ S･, House, L H･) Fehler, M･ C･, 1997･ Detailed Joint Structure in a Geothermal

ReservOir from Studies of Induced Micro Earthquakes Clusters. ). Geophys. Res･, 102, ll,

745-ll, 763.

Poupinet, G･, GlangeaudH F･? Cote, 0･, 1982･ P-time Delay Measurement of a Doublet of Micro

Earthquakes, in: Proc. IEEE, ICASSP82, 1516- 1519.

Saito, H. and Hayashi, K., 2002. "Hydraulic Properties Evaluation from Stoneley Wave Attenuation by Using Permeable Fracture Zone Modeln, Geother･ Resour･ Coun･ Trans･, 26,

319-323.

Takahashi, H. and Hashida, T., 1992. "New Project for Hot Wet Rock Geothemal Reservoir Design Concept", Proc. 17th Stanford mark-hop on Geothermal Reservoir Engineering, StanfoJ4 39-44.

ー25-

Earthls surface

FigureII-2. Schematic representation of the set up･ Crack waves were observed by hydrophone during drilling of the well EE-5･ (Higashi-Hachimantai)

-27- -9Z-(叩ut!tH! tPq-!qS嘘!H)

･19POt叩〇W lu9S9Jd 9ql uOq p9tr!t!lqO S9Pu9TtbaJJ 耳e9d p911!tn叩3 1hOqS SJhOJJV ●uO!lt!Z!JaPt!112q〇 JOJ

p9STt 9Jt! (P9PJP) S9Ptl9nbaJJ平9d ･1-d tt9Jh JO 9JnSS9Jd pで9tm9瓜叩S! LuJ I;13耳tl9Jh 9ttl JO Su!ttpp知pnp p9JuaSqO叩P Pl叩9tP JO叩39ds 9Pnl!tduc JO 9tdtmxヨ ●C-ⅠⅠ9Jtl軸

(zH)血ttanb9Jd oo I OS O9      0ウ      0て ■V,K

t!JMllZ-a ll.ll.1.ll

`≠ t!JMO●1-a I.Iレl

A

m

p

t

i

t

u

d

e

●

■ ■

■

■

0       1      2       3

Wellhead pressure (MPa)

FigureII-4. Variation of estimated radius with wellhead pressure･ (Higashi -Hachim antai)

-29-雷 }   ■     ●

-OS-(!t!lUt! u!tPt!H一円S12g!H) ･9JnSS91d pc9qtt弧tP!JMJrLIJ9dtZ P叩tH!lS9 JO uO叩Pt!^ I;-ⅠⅠ9JnB!d

(t!dM) 9JnSSaJd pで9ⅦP瓜 E Z      1      0 ● ナO IXO'乞

%

a

ト｡-Ⅹ.･ケ旨

(

且

ヶO tXO●⊆

-te-(!t!1tn!uI叩Pt!H-!qS嘩H) '9JnSS9Jd

pt"ttttaJh WJhで9 SS9Ⅶ!lS t甲印31u! P叩tH!1S9 JO uO!l叩t!^ ･9-ⅠⅠ9JnS!d

(甘dM) 9JnSSaJd pt!9Ⅶ19瓜 e Z I 0

2 0 -2 -4 -6 .6   1   _2   0   2   4   6 East仲m)

Figure I1-7･Lecations of AE observation wells in Australian HDR

丘eld. Ha♭l is the injection well (Habanero#1)･ A mark ▲

means the location of epicenter. (Australian HDR site) -32-書 名 h O N A V 4

0.0 -4.Ox1 0･5

lflll EW

_L止血山LLLb.山LLJAl_.山山山Lh.[山..L....山.i.L,h"山.一山⊥JJJ_.._‥. lJJ山山山LJLJ-山山上LLJLLb 啖ﾂ 叩1lm叩q''rq.叩1rt"rrVlyTl-｢ ー'-ー ■ー tllll lllll NS __..山血山血一山hJ.止山止血山山.i.｣.山.｣山LJl"J...__ 剿ﾃ "lnl山山山tidLiA山山■山▲一▲】- 叩叩VTFrVmLm'"叩rrmlWTrTywTT■"'T叩rTTql-1T'■TIPrPrr｢ lllll l 免ﾆﾆﾆﾂT 佇ﾉ Y Xﾈﾈ 謦 萃韃"ﾂ"粨 ｲ よ亥$､｢ﾂﾂ褪粨TBｵ WWYm.TーrrrLL【叩叩叩■町1lW一◆l¶ ■''Tr-■ ' T-"丁 lllll 0     50    100    150    200    250    300 Time (sec)

Figure II-8. Wave forms of the event observed at well WA4. These are

used fわr characterization (Australian HDR site)

-33-0 I X 0 4 5             0 ( s J u 3 ) A l ! 3 0 t a A 5         5 0   0 1     1 X X 0 0 4   4 0         . 5   . 5 0 .     t o t o X X O 0 4   4

llll ■■■■EW

/.-1､.-.-..I._.-WA2

･rヽ一._..､-.-.-WA4 ､■ヽ ′ ヽ llll.∫..′ヽ ''''NS 一一.′■_､.′.＼-WA2 ヽ..,...--一一一WA4 ●ー◆､/ヽ. ヽ一 llll●′●一■..- ■■■■uD '-､′'､=wwT. ､./■ ■一一 ∼ ●ヽ･● ll■l 0     2     4     6     8    10 Frcquency (Ik)

Figure II-9･ Examples of amplitude spectra of the field data

observed at wells WA2 and WA4. Peak frequencies

(circled) are used for characterization･ Peaks between 3 and 6 Hzand over 7 Hz are not chosen because

spectra in these ranges reflect the effect of natural

frequency of seismographs and background･Anows

show calculated peak frequencies obtained from the present crack model･ (Australian HDR site)

-34-( 3 1 8 S b q ! q J t I ) t q D a d s J a J h O J

Appendix

FornulatioA Bor Axisymmetric Deformation

Motion of FIuid

Let us discuss the dynamics of the fluid in the crack･･ It is assumed that the

amplitude of oscillation of the crack su血ce is small enough00mpared to the initial

aperture of the crackand the flow is symmetric withrespect t the x3-Plane･ It is

assumed that the change of the fluid density during the oscillation is also negligibly small･ Then the equations of the motion and continuity are glVen by

告ニー器･言(箸写)

告雷･霊(ru,/)･p,i - o

(1)

(2)

where p, , 77, b are the density, viscosity and modulus of volume expansion ( or

bulk modulus) of the fluid, respectively. The functions u,I , u,I are the velocities of

thefluidflow in the crack in the radial and x3-direction･ On the crack surface, we

have,from the continuity,

札→喜一｡ -札→言.0 +

I ∂血3

2 ∂/

(3)

where u3f is the velocity of the fluid in the rock･ Regarding thefluidflow in the rock,

let us assume that flow is ID in the x3 -direction and accept Darcy's law･ Then we have

zL;,= ko ape Tl ar3

i(mopJ･)･i(p,･u;･) - o

Al (4) (5)

where mo and ko are the porosity and permeability of the rock, respectively and Ps is the

pore pressure in the rock･ For the boundary condition, we set as follows:

p=o at r=0,      (6)

since a well is crossing the crack at the centerand the well volume is sufficiently higher

than the crack volume. If there is no well connected, the fわllowing boundary condition

should be employed instead of Equation(6): 些=o al r=0.

dr

Regarding the boundary condition at the crack periphery, we set

I

dFd/,22 u,1,.a_｡血3 -普r一LZ ll

A･1う0

(7)

(8)

These are the basic equations and the boundary conditions･ Now we move

into the hplace image space, transform every equation and perform the integrationwith

respect to x3 ･ Then finally wearrive at the followlng differential equation to

determine the fluid pressure in the crack:

完結anh(写) -.)g

境il(,･k)(孟tanh(写)llhttanh2(写)]雷 (9,

･〔

空室.

b2

which is accompanied by the boundary conditions glVem by 声-o(ord声/dr-0) at r-0.

aF L n-2才肝

･-∴ itanh(写)

arl,=o ヱtanh(型l_I

W, I :;:h｡

wherep is the variable of Laplace transform and

n-2 -声等

A2

(10)

(ll)

Boundary Integral Equation

._●

h addition to the applied no-al stress α 33, We must Care the pressure of

瓜uid and the surface traction due to par(ial contact between asperities on the two

surfaces of the crack･ Thus the boundary condition is expressed in the form such that

5JA.)→.0 - kAi1, -P-･6-;, (on the crack surface),  (13)

where k isthe interfacial stifhess due to the contact, Au-3 is the displacement gap and

p is the nuid pressure･ The present problem we are now formulating lS aXISymmetric,

so that the field quantities expressed by the cylindrical coordinate system are

independent of the circumferential coordinate･ By performing the integrations with respect to the circumferential coordinate, we finally arrived at thefollowlng eXPreSSions･

The boundary condition (Equation(13))is expressed as

X-(R,S)+VIR.S)-6(RmR,S)-i(R,S) (0<R<l) ..

Here and in thefollowlng We use thefollowlng nOn-dimensional notations:

X,=互,Y,=左.R=工,S=里.Z-呈 a a a a a 6(R)-!k(r)･ S-貰･ G--ay-I P-也 〃 ､′■  cT一   ′'一  αCr

QIR.S)=蓮如適. g-(R,S)=幽

2 a2 '01 C, 5-" (r･凡→.0 X-(R.S) = all wl..(X.S) = ⊆麺適 a2 α/∫ V-(R.S) = ⊆塾適 aFL (14) (15)

where D- (X.･-E.I)(X.1-Ei)and.E - (X1-51)2.(X2-52)2 ･ D represents the

distance between the two points in the 3D space and E is the distance between the two

points on the crack surface. In Equation(14), ￠ and y' are the displacement gap

and the fluid pressure, both of which are unknown to be determined･ The function x

is the stress on the crack surface and is glVen by

I-(R･S) - i(1 -i)iprf.灘dC ･去prf.禦dE･I

(i(. -.i)2万)pfL-). 4-(C,S,･.nlc - RL･ dc

(E(C)-.)一三百毒]dc ･･

-R)2l一丁′ L'2(6+R)2 G-2 7r

(帰)2万)i QlC･StH(C,三･nlb12 - 1中･

一志l^: QlY･S)･ X-･(X･Y･ shA(Y)

The function x, in the last term oftheright hand side of Equation (16) is given by

I-,(x.y.s)

- (K2 -2ア写(e5z l.)･4(K2 l l除去)e0-I -(去喜一去〕)

-tl(K2 -2劇悠eo-l,r -(1-i))

･4(K2 -2%(eoIz -feolz,･ -(1 -fH.一去f-Z) -4(K2 -56te0-a -吉e0-2′･ -(.一志)-(. -i)G-Z

2L K

(1 - i)(G-zf )

-36g(eo-I -ieG-I,･ -(. 1三)-(. -i)G-Z甘f)(GIzf

甘さ)(G-zy) +36g(eG-I -eG-I/. -(.一三)G-Z -i(. -i)(G-zy

ーi(. - i)(G-zy一去(1 - i)(G-zr)]

. (17)

The functions E(C)and K(C) are the elliptic integrals of the first and second kinds, respectively, where c is defhed by

2J哀

C= 一一

6+R

(18)

It should be noted that K used here is equal to the inverse of the ratio k used

in Part I (see Equation (I-14)),i.e., K -CL /CT･ The ratio is 2 when the Poisson's

ratiois 1/3.

Numerical Procedure

Equations (9)-(12) are rewritten by using the non-dimensional notations

introduced in Equation (15) asfollows:

(aanh(Z) -.厚

[(1 ･剥stanh(ZH ･ktanh2(Z)]慧

mo p/pal P p,bdつ V-IR=｡ = 0 空l = eL sh-2p P. 朗IR-.空Ianh hd I ∫ h-2 -万･才

(計

W-, 12㍍

一票sh-2pQ-(19) (20) (21) (21)

Thus it is readily understood that the problem is reduced to solving the integral equation

(Equation (14))and the differential equation (Equation (19)) subjected to the boundary

conditions given by Equations (20) and (21)･ Regarding the differential equation, we

employ FDM･ The singular integralequation is solved by a collocation technique

based on Gauss-Chebyshev qradrature formula･ By taking account of the asymptotic behavior of the displacement gap in the vicinity of the Crack periphery, we set such that

QIR.S) - √=京王A-. (shl. (R)      (22)

l=l･

where A-k is unknownand Uk(R) is Chebyshev polynomial of the second kindsthat

is defhed by

Uk (R) - sin(kcos-lR R)/sin(k cos-I R).       (23)

Schematically speaking, the problem is finally reduced to solving the simultaneous linear algebraic equations for Ak , i･e･

(24)

(25)

(26)

(27)

By substituting Equation (25) into Equation (22) and apply IAplace inverse transform, weget

dR, T) -正平差if-'iニ鮮Tds ･ U2k-I (R) '28'

Here, T is non-dimensional time (7 - CTt/a ) and b is a real number lager than the

real parts of the zero points of the matrix A(S). The zero points are denoted as

aJ･,i･e･, A (all)-0･ The complex conjugateof aj is alsothezeropoint･ Thenwe

arrive at thefollowlng schematic representation of the displacement gap:

･R･ T, -若戸善2Re(卦J, ) ･ U2k-I (R,] (29,

The above equation indicates that the imaglnary parts Of αJ is the elgen angular

frequencies of the fundamental modes of oscillation of the surface of the fluid-filled

crack. A6 〃  〃 一 d t I B ∼ " + 十 + lQ.･l触三 宮 -一 b c r . ･ ･ ･ t 一           1 l 一 一 礼 I J Q g S ' ' + l e i 十 一 ‰ J g 〟 ｣ E g g l 恥