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(1)æ~>Æ·~ã²æ. òY. Vol. 9, No. 1, pp. 79~86, 2004. 4>ê> Ö;j * îz +»~ 'Ï. j÷ . æî¶öö æî~ãÒ¦ Application of the Homogenization Analysis to Calculation of a Permeability Coefficient Byung-Gon Chae Geological and Environmental Hazards Division, Korea Institute of Geoscience and Mineral Resources. ABSTRACT Hydraulic conductivity along rock fracture is mainly dependent on fracture geometries such as orientation, aperture, roughness and connectivity. Therefore, it needs to consider fracture geometries sufficiently on a fracture model for a numerical analysis to calculate permeability coefficient in a fracture. This study performed new type of numerical analysis using a homogenization analysis method to calculate permeability coefficient accurately along single fractures with several fracture models that were considered fracture geometries as much as possible. First of all, fracture roughness and aperture variation due to normal stress applied on a fracture were directly measured under a confocal laser scaning microscope (CLSM). The acquired geometric data were used as input data to construct fracture models for the homogenization analysis (HA). Using the constructed fracture models, the homogenization analysis method can compute permeability coefficient with consideration of material properties both in microscale and in macroscale. The HA is a new type of perturbation theory developed to characterize the behavior of a micro inhomogeneous material with a periodic microstructure. It calculates micro scale permeability coefficient at homogeneous microscale, and then, computes a homogenized permeability coefficient (C-permeability coefficient) at macro scale. Therefore, it is possible to analyze accurate characteristics of permeability reflected with local effect of facture geometry. Several computations of the HA were conducted to prove validity of the HA results compared with the empirical equations of permeability in the previous studies using the constructed 2-D fracture models. The model can be classified into a parallel plate model that has fracture roughness and identical aperture along a fracture. According to the computation results, the conventional C-permeability coefficients have values in the range of the same order or difference of one order from the permeability coefficients calculated by an empirical equation. It means that the HA result is valid to calculate permeability coefficient along a fracture. However, it should be noted that C-permeability coefficient is more accurate result than the preexisting equations of permeability calculation, because the HA considers permeability characteristics of locally inhomogeneous fracture geometries and material properties both in microscale and macroscale. Key words : homogenization analysis, roughness, aperture, confocal laser scanning microsocpe (CLSM), C-permeability coefficient. º £ ^. zC Ú j V >Ò*êêº ~ V~' º², ¯ OË, *, V Ò ç^ Öêö " ²ÖB. . V¢B, Ú R>ê>¢ ;{~² êÖ~V *Bº f ?f V~ º² j & êÖÎö >' j º& ® . öBº V~·çj & ;{® >' ÎöB V >~C"º îÚ O» îz C»(homogenization analysis method)j Ï~ j V R>ê>¢ ~V * >~Cj >¯ *Corresponding author : [email protected] : 2004. 1. 16 : 2004. 3. 2 : 2004. 6. 30. ö%>¢ î~ # Æ~. ²Òß¢ Hæ. 79.

(2) j÷. 80. ~& . b&, .6 .& Êº *ã(Confocal Laser Scanning Microscope)j Ï~ zCò~ êf ö & >ç{»K~ æzö V * æzïj ç7 G;~, f ? ³ ¶òº Î Ò*j * «K¶ò ÒÏ>î . Ò*B Îj Æ& îz C»f Î(microscale) îßW" Î (macroscale) îßWj ÿö J~ R>ê>¢ êÖ > ®º © . ¯, îz C»f "V' ^ (microstructure)¢ <º ² ®î bî~ ÿßWj «~V * BBB îÚ ;~ Sÿ(perturbation) . º î ÎöB R>ßWj êÖ ê, ÎöB~ îz R>ê>¢ êÖ~² B . æ, O»f V~·ç~ ¦' 'Ëj J R>ßWj ;{® C > ® . îz»j Ï R> ê> Ö;Ö"¢ V öB Bn ãþ" jv~ æWj ¦Ã~V * *F 2Nö Îj Ï R>ê> êÖj >¯~& . Îf V(roughness)¢ >'~ ÿ¢ *j ï¯6 Îj &; ~& . êÖÖ"ö ~~, îz +»ö ~ êÖ C-R>ê>º ÚR>þö ~ R>ê>f ?f º*~ 8j &æ¾ 10 ;ê~ N¢ , êÖÖ"º æ~ " > ® . ¾, îz +»f ¦ 'b ®î V~·ç" bîß9 Îf Îö* Îv J>æ, ß9j ;{® r ® j ãÖ Vö BnB ãþ ö ~ êÖÖ" îz +»~ Ö"& R ;{j "Ï~¢ . *Ú : îz +, V, *, .6 .& Êº *ã, C-R>ê> −1. 1.. * V. Æ· ?f Ö >î &ç~ FÚFÿ ÎÒVFj :ûb Bê ¾ z>ö*~ FÚFÿ ß9j ;{® ÎÒ~V * ÿn ôf \& ê>î . ~ ;f î~ ß9ö "~ V\¢ \ª~ ² 5&æ ¾æâ > ® . b&, ï¯6 ; "æ~ îf ®R9 ãÖ¢ J FÚF ÿ ÎÒBL . Îö* R>Nf *~ âßö jf º âß»j êÂ~& . v®º ï ¯6 ;~ j VB~ ê¢ J ®ïê ;f ®R>W îj Î BL B>î . ÎöBº *Ú 'ö & 7/¦ª~ jNj J~&b, ê¢ J«~º îVKj J~& . ^®º FÚFÿ ÎÒ ®ïê ~ V~·çj ;{® >' > ìrö V~ ï¯6 Îj & W î BL ê«>î . º ¦ªj ï¯ W î &;~ ¢ Û R>ßWj C®J Î . Îj Æ& "æ î R>W ãÖ¢ ÎÒ~V * 7 W î Î Bn> Vê ~& . ¾ "öº B zC~ V~· çj & >'~ ¢ ;{~² ÎÒ~V * K. " þ Î ê¯ 7, ¢ :ûb * ËÎö 'Ï~ ®. . zC Ú j V >Ò*êêº j W~º V ~' º², ¯ OË, *, V Ò ç^ Öê ö " ²ÖB . V¢B, >Ò*êê¢ ;{~² êÖ~ V *Bº f ?f V~ º² j & êÖÎö 1-6). 1,7-15). 16-18). 19). 3,10,19-27). Journal of KoSSGE Vol. 9, No. 1, pp. 79~86, 2004. >' jº& ® . zC j V >Ò*êêº ß ® V~º² 7 *~ æzö Ö "6~² ² ÖN C&æB *~ 'Ëj J~V * *F :f ?f ï¯6 Îj &; âß»j ÒÏ~&b ¾, "ö âß»f >~ ö ~ ®ïê j V FÚFÿj ;{~² C > ìr ¾æÒ. . æ, z> j &çb ®ïê ;f ö V *æz¢ ;&~² G;~ ¢ Ï~ ;{ R>ê>¢ êÖ~º © jº~ . öBº V~·çj & ;{® >' ÎöB V >~C"º îÚ O»j Ï~ j V R>ê>¢ ~V * >~Cj >¯~& . ;{ ~ êf *G;j *~ " þ öB ÒÏ®~ © z ¸f çê~ .6 .& Êº *ã(Confocal Laser Scanning Microscope) j Ï~ zCò~ ê¢ G;~& . $, ö & ¢»{»K~ æzö V * æzïj .6 .& Êº *ã ~öB ç7 G;~& . f ? ³ V~¶ò¢ Ï~ Îj Ò *~& . Îj Ï >Ò*êê êÖj *B º Î(microscale)~ îßW" Î(macroscale) ~ ßWj ÿö J~ >Ò*êê¢ êÖ > ®º îz C(homogenization analysis)j Ï~& . î z C»f Sÿ(perturbation) j 'Ï ®î OW î ÚöBê î OWj º .Î~ ^ ßWj ¾æÚº ²O;" Î .Î~ ßWj ~º &O;j ê« Î 5 Î~ ç^ êB R>ßWj * . ¢ Û Î~ 12,20,24). 28). 29).

(3) 4>ê> Ö;j * îz +»~ 'Ï. ¢ ;{® >' R>ßW C &Ë~ . 2.. 1. 1. 2. 2. 2 ε. in Ωεf ,. in Ωεf ε. Vi =0. on. (1) ∂Ωεf. (2). VB V : *6W ηj &æº ³êÇV, P : {K F : bÚK(body force) vector Ω : *Ú ²êöB~ FÚFÿ '(δΩ : ãê) B r" ?f 6"*Bj ê« > ®º, ε i. ε. i. εf. α i. α. α. 30-31). ε. α. α i. îz C»f Sÿj 'Ï ©bB, "V' ^¢ &æº ®î bî~ ÿj ~V * BB>î (Fig. 1) . öBº V' j : ûb ÚöB~ FÚFÿ ^B¢ C~V * îz»j wÏ~& . îz C»f Îf ÎöB~ ßWj ÿ ö êÖ~æ v «~~ ²Úê¢ &ê . V¢B, ÎöB~ ¦ ²ê¢ y, Î~ *Ú ² êº x ;~~, y = x/ε~ &ê¢ &ê . VB εf 2¢zV(micro parameter), Fig. 1ö ¾æÞ : f ? ÎöB f(cell)~ V& (εY , εY ) >, Î~ * f~ Vº(Y , Y )& B . îz»f b& r" ?f j{»W Navier-Stokes O; öB¦V · . ∂V --------=0 ∂ xi. VB V (x, y)f P (x, y)(=0, 1, Ã)º * f Y~ V¢ <º Y-"V>B V (x) = V (x, y + Y), P (x, y) = P (x, y + Y)f ? . *F & y = x/ε ~ &ê¢ &ææ r" ? ªj :æ > ® . α i. îz +». ε ∂ Vi P ∂--------+ - +F =0 η --------------∂ xi ∂ xk ∂ xk i. 81. ∂ ∂ 1∂ ------- ⇒ ------- + --- ------∂ xi ∂ x i ε ∂ yi. (4). (3)" (4)¢ (1)ö &«~ ε 0b ~ r" ?f &ê êÂB . ∂P0 0 0 –1 ε –term : --------- =0 ⇒ P ( x, y )=P ( x, y) ∂yi. in Yf. (5). 2 0. ∂ Vi ∂ P0 ∂ P1 0 ε –term : – ---------+ η ---------------- = --------- –Fi ∂ yi ∂yk ∂yk ∂xi. in Yf. (6). f FÒ~² (3)f j¾f ? FêB . 0. ∂ Vi ε –term : ---------=0 ∂yi 1. 0. (7). in Yf 1. ∂ Vi ∂ Vi 2 ε –term : --------- + ---------=0 ∂xi ∂yj. (8). in Yf. FÚf Ú~ ãê f (9)f ? *B . 0. 1. Vi =0,. on Γ. Vi =0,K. (9). Þ, (6)ö æ>ªÒ»j 'Ï~ r" ? ;Ò > ® . 0 0  ∂P (x ) k Vi = Fk(x ) – ---------------- νi ( y ) ∂xk  . εf. ε. 2 0. 3 1. ε. 0. 1. Vi (x )= ε Vi ( x, y)+ ε Vi ( x, y )+K, P (x )=P (x, y )+ε P ( x, y )+K. 0 ∂P (x ) k 1  P = Fk(x ) – ---------------- p ( y) ∂xk  . (10). VB, ν (k = 1, 2, 3) : ßW³ê, p (k = 1, 2, 3) : ßW{K , (6)f yö & ÞªO;b æ~B . k i. (3). k. 2 k. ∂ νi ∂p -+δ =0 –------- + η ------------∂ yi ∂yj∂ yj ik. in Yf. (11). f FÒ O»b îï» ( (7)), FÚ-Ú ãê¦ ( (9)), Ò "V' ãêf r" ? *B . k. ∂νi -------- =0 ∂yi k. in Yf k. νi ( y)= νi ( y + Y ),. Fig. 1. Schematic concept of the homogenization analysis.. k. νi =0 k. on Γ k. p ( y)=p (y + Y ). (12) on ∂Yf. (13). VB, (11)-(13)f O;(micro scale equations; Journal of KoSSGE Vol. 9, No. 1, pp. 79~86, 2004.

(4) j÷. 82. ¢ . ¢ :ûb (11)ö & ïz¢ .. MiSE). 0 ∂P V˜ i =K ji Fj – ------- ,  ∂xj. j 1 j Kji=νi= -------- ∫ νi dy YY. (14). f. VB, V˜ : *f (|Y|) ÚöB ïbÚ³ê K : îz R>ê>(HA-Permeability) ÿ¢ ïz¢ (8)ö 'Ï~, V "V & ê v ® ²B . V¢B, (15)f ?f O;(macro scale equation; MaSE) êÂ>, ¢ îz Lª O;(HA-flow equation) ¢ . 0 i. ij. 1 i. 0. ∂ Vi --------- =0 ∂ xi. 0  ∂P  ∂ or ----- KjiFj – --------- =0 ∂xj  ∂x . in Ω. (15). {K P f ³ê V (3)ö ~ "Ò'b r" ? êÖB . i. ε. 2 0. Vi (x ) ≅ ε Vi ( x, y ),. ε. 0. P (x) ≅ P (x). (16). *öB J« ";j º£~ r" ? . Ñ, (11)-(13)ö ~º O;j ¦, ν f p ¢ r, (14)öB K ö ~ Darcy ê>¢ Ö; . Ò, (15)~ O;j ¦ÚB {K(macro pressure)¢ . ¢ (16)ö &«~ «'b. B {KË(pressure field)" ³êË(velocity field)j Þ . ¢>'bº Darcy »j j¾f ? *~ z . k i. k. ij. ∂H V˜ i′= –Kij′ ------- ; ∂xj. P H= ------ + ζ ρg. (17). VB, V˜ ′ : ï ³ê, H : * >v, P : {K>v ζ : *~>v, g : 7K&³ê ρ : b~ îï(j{»Wæ ¢;) (15)-(17)j jv~ r" ?f &w&ê& ®bæ, i. ε 2 0 V˜ i′=V˜ i ≅ ε V˜ i. (18). R>ê> K f Ûç'b ÒÏ N C-permeability K ' Òöº j¾f ?f &ê¢ &öj r > ® .. HA-. ij. ij. 2. Kij′= ε ρgK ij. 3.. V(roughness)º >'~&b¾ ·ã B ï ¯ ï¯6 Îj &;~& . ï¯6 Î &; Fº îz»~ êÖÖ"f âß»(cubic law)ö :û j z V~ ãþ" jv~V * © . 2Nö Îf .6 .& Êº *ã(confocal laser scanning microscope; CLSM)j Ï~ ~ V(roughness)f *(aperture)¢ ;& G; Ö"¢ Æ& *~& . b&, CLSM &Vj Û · ã~ V ¶ò¢ ;ï'b ç7 G;~& . ³ ¶òº ³ Òö æ~(fast Fourier transform)j Ï Ê¿Þ" ªCj ê, low pass filter¢ Ï~ Çrj B~& . ¢ Òö æ~(inverse Fourier transform)j Û Çr BB V ¶ò Ò*~&. . Ò*B ¶òº 1 mm *Ïb þ2ç ê 2 Nö Î~ V «K¶ò ÒÏ~& . ï¯6 Î~ «K¶ò Ï $ ~¾~ ¶òº ï *(mean aperture) . CLSMö ËOB ß> B·B {»Ë~¢ Ï~ ö 5ê~ >çwK j &~, ÿö CLSMb ÿ¢ òö & *~ æz·çj ³'b G;~& . {Kêê ' G ;æ6öB~ * 8j Æ& òê ï *j ~ &, ß® Zimmerman" Bodvarsson(1996) Bn (20)ö ~ >Ò *(hydraulic aperture)¢ ê, ¢ ï~& (Table 3 in Chae et al., 2003) . -² ³ v «~~ ï *j Ï~ V¢ J Îj *~& (Fig. 2). 28). 29). 20). 29). s 2 3 bh= ⟨ b⟩ 1 – 1.5  -------  ⟨ b⟩ . (20). VB, b : >Ò' * b : >Ò' * s : R&ÞN Þ, CLSMöB & >çwK" ÿ¢ V~ /{ (confining pressure)j òö &~B j V R> Wj 2k~¶ ÚR>þj ~& . ¯, ö & >çwK" ÿ¢ V~ /{ &ææ * ~ Væzê CLSMöB G; 8" ÿ¢~² æz h. 29). (19). ï¯6 Îj Ï îz +» ¦Ã. îz»j Ï R>ê> Ö;~ æWj ¦Ã~V * 2Nö Îj Ï~ êÖ~& . Îf Journal of KoSSGE Vol. 9, No. 1, pp. 79~86, 2004. Fig. 2. A parallel plate fracture model used for the HA simulations (10 MPa of GRA). Exaggerated 100 times in vertical direction..

(5) 4>ê> Ö;j * îz +»~ 'Ï. 83. Table 1. Results of the laboratory permeability test and hydraulic conductivity Specimen. GRA. GRB. GRC. GRD. GRE. GRF. Pconf. (MPa) 10 15 20 10 15 20 10 15 20 10 15 20 10 15 20 10 15 20. Time (sec) 48.88 71.50 104.15 19.47 22.45 24.25 4.00 4.84 5.81 2.00 2.14 2.21 4.22 10.94 79.75 165.66 1,182.68 4,032.00. j 6n~, * Vf R>N(flow rate)"~ &ê¢ 2k~¶ ~& . Table 1f * G;Ö"f ÚR> þöB G; R>N" ¢ Æ& Darcy »j Ï R>ê> êÖ Ö"¢ ¾æÞ © . ÎöB Î ãêf V f P ~ "VW j ò¢ ~, çöBº ³ê& 0Ú¢ . K i. k. Flow rate (cm3/sec) 1.023 0.699 0.480 2.568 2.227 2.062 12.500 10.331 8.606 25.000 23.364 22.624 11.848 4.570 0.627 0.121 0.017 0.005. Aperture (cm) 0.010 0.009 0.009 0.018 0.017 0.016 0.031 0.031 0.030 0.043 0.042 0.038 0.026 0.026 0.025 0.011 0.011 0.011. Hydraulic conduct (cm/sec) 2.780E-04 1.367E-04 6.687E-05 9.430E-04 7.616E-04 6.821E-04 1.311E-02 9.066E-03 6.358E-03 3.845E-02 3.424E-02 3.480E-02 1.382E-02 2.122E-03 4.109E-05 8.464E-06 1.699E-07 1.494E-08. b~ Lªf Ú¦öB B~æ zC~ îf ® R>W¢ &; . >~Cf Nê 300 K¢r¢ &;~ >¯~& . r b~ 6W ηº 0.8Ü10 PaÁsec, &ê ρº 0.99651g cm . Table 2f Fig 3f *F V~ G;¶ò ¢ Ï Îj W~ ¢ Æ& îz C −3. −3. Table 2. Permeability values calculated by Darcy's empirical equations (Km*, Kh*) and the conventional C-permeability (Km', Kh') Specimen GRA. GRB. GRC. GRD. GRE. GRF. Pconf. (MPa) 10 15 20 10 15 20 10 15 20 10 15 20 10 15 20 10 15 20. Km* (cm/sec) 2.780E-04 1.367E-04 6.687E-05 9.430E-04 7.616E-04 6.821E-04 1.311E-02 9.066E-03 6.358E-03 3.845E-02 3.424E-02 3.480E-02 1.382E-02 2.122E-03 4.109E-05 8.464E-06 1.699E-07 1.494E-08. Kh* (cm/sec) 2.845E-04 1.404E-04 6.884E-05 9.630E-04 7.794E-04 6.970E-04 1.401E-02 9.718E-03 6.830E-03 3.872E-02 3.446E-02 3.533E-02 1.390E-02 2.134E-03 4.132E-05 8.541E-06 1.715E-07 1.508E-08. Km' (cm/sec) 9.192E-04 8.722E-04 8.105E-04 3.332E-03 2.891E-03 2.646E-03 7.409E-03 7.223E-03 7.076E-03 1.803E-02 1.735E-02 1.480E-02 6.909E-03 6.497E-03 6.203E-03 1.137E-03 1.205E-03 1.156E-03. Kh' (cm/sec) 9.232E-04 8.271E-04 7.644E-04 3.195E-03 2.754E-03 2.538E-03 6.478E-03 6.292E-03 6.125E-03 1.784E-02 1.715E-02 1.431E-02 6.831E-03 6.419E-03 6.066E-03 1.245E-03 1.186E-03 1.137E-03. Journal of KoSSGE Vol. 9, No. 1, pp. 79~86, 2004.

(6) 84. j÷. Fig. 3. Comparison of the permeability coefficients acquired by the laboratory water injection test (circle) with the C-permeability coefficients (triangle) calculated by the HA. Journal of KoSSGE Vol. 9, No. 1, pp. 79~86, 2004.

(7) 4>ê> Ö;j * îz +»~ 'Ï. »ö ~ êÖ R>ê> . VB K ' " K ' º CLSMb G; bÒ' *(mechanical aperture) bm " (20)ö ~ êÖ >Ò' * b ¢ Ï~ îz»b êÖ R>ê>, K *" K *º ÚR> þöB G; R>N" b " b ¢ Darcy ãþö &«~ R>ê> . îz C»ö ~ êÖ C-R>ê>º ÚR> þö ~ R>ê>f ?f º*~ 8j &æ¾ 10 ;ê~ N¢ , êÖÖ"º æ~ " > ® . C-R>ê>º ï¯6j Û b~ Lªæ, âß»öB "Ë~º ' R>ê>f *"~ & ê¢ &Ú Vº ©j r > ® . ¯, &¦ª~ ò. f {Kö V * 8" R>ê> 8 Òö K b ~ &ê¢ Fæ~B jf'b æz . ¢ Û öB ÒÏ îz C»f zC j V R> W Ö;ö Î"'b Ï > ®rj r > ® . m. h. h. m. m. h. h. −1. 2. 4.. Æ ~. öB ÚR>þöB R>ê>º âß»~ ' &ê¢ Væ pº . º ÿn ¶ "Ë~ Chae et al.(2003) ~ Ö"öB ¾æÂ &, ¶Úº ï¯~æ p ®ïê ; ¢ &æ ®bæ ï¯6j &; âß»~ ' &êf N& ®V r^ . ¯, öB ÒÏ f ®ïê æ ÚR>þj Û R>ê>º ~ Vf ¦'b *~ ' Ëj

(8) Ö" . $, Fig. 3~ C-R>ê>º êÏ® ö âß»~ ' &ê¢ jã~² Væº pº . Fig. 2~ Îf ÿ¢ *j &æ ®Ú ï¯6 Î *" >º ®æò, j &ê V¢

(9) ~ ®. . *Û'b ï¯6 ÎöBº ö V& ì º Þï ~ ç¢ &;~& . ¾, öBº V¢ &ê ï¯6 Îj &;~&bæ, FÚ Fÿ "~ ãê¦öBº Â~(turbulent flow) ~ 'Ëj Aj > ® . F r^ö îz C »ö ~ C-R>ê>º âß»~ ' &ê¢ Æ&® Væ p £*~ N& B ©b 'B. . f ?f 6j 6n~ âß»f B ¶êö Ò~º j V R>ßWj ;{® >'~V ÚJ æj r > ® . öB Bn îz C»f ~ V~·ç Fig. 3. 29). 85. j ;{~² >' > ®rj "Ï jº& ® . ¦ 'b ®î V~·ç" bîßW Îf ÎöB Îv J>æ, ~ ¦' V~·ç " bîßWj ;{® r ®j ãÖ, îz C»f Vö BnB ãþ ö ~ êÖÖ" R ;{ Ö"¢ êÂ â > ® . V¢B, B ¶êöB >÷ ' j V R>ßW ï&ö îz C»f Î"' ;{ Ö"¢ &^R > ®j © . 5.. Ö V. ~ V~·çj ;{® >'~ j V R> ßWj 2k~¶ îz C»j ê«~ R>ê>¢ Ö;~& . îz C»f Sÿj 'Ï ©b B, "V' ^¢ &æº ®î bî~ ÿj ~V * BB>î . îz»f Îf Îö B~ ßWj ÿö êÖ~æ, ®î V~·ç" bî ßWj <º öB ;{ CÖ"¢ êÂ > ® . îz»j Ï R>ê> Ö;~ æWj ¦Ã~V * 2Nö Îj Ï R>ê> êÖj >¯~& . Îf V(roughness)º >'~&b¾ ·ã B ï¯ ï¯6 Îj &;~&b, Fº îz»~ êÖÖ"f âß»(cubic law)ö :ûj z V ~ ãþ" jv~V * © . îz C»ö ~ êÖ C-R>ê>º ÚR>þö ~ R> ê>f ?f º*~ 8j &æ¾ 10 ;ê~ N¢ , êÖÖ"º æ~ " > ® . $, C-R >ê>º âß»öB "Ë~º ' R>ê>f * "~ &ê¢ &Ú Vº ©j r > ®Ú, zC j V R>W Ö;ö îz C»j Î"'b Ï > ®º ©b 'B . , îz C»f ¦ 'b ®î V~·ç" bîßW Îf ÎöB Îv J>æ, Chae et al.(2004) f Chae et al.(2003) öB ÒÏ O» j Û ß Wj ;{® r ®j ãÖ Vö BnB ãþ ö ~ êÖÖ" îz C»~ Ö"& R ;{ j "Ï~¢ . −1. 28). 29). Ò Ò º 21^V *ÚBBÒë >¶ö~ æ ³' {VFBBÒë~ jæö("B®^ 3-2-1)ö ~ >¯>îÛî . Journal of KoSSGE Vol. 9, No. 1, pp. 79~86, 2004.

(10) j÷. 86. ^^ò 1. Rehshaw, C.E., “On the relationship between mechanical and hydraulic apertures in rough-walled fractures”, Jour. Geophys. Res., 100(B12), pp. 24629-24636 (1995). 2. Tsang, Y.W., and Witherspoon, P.A., “Hydromechanical behavior of a deformable rock fracture subject to normal stress”, Jour. Geophys. Res., 86(B10), pp. 9287-9298 (1981). 3. Kranz, R.L., Frankel, A., and Engelder, T., “The permeability of whole and jointed Barre Granite”, Eos Trans. AGU, 58(12), pp. 1229 (1979). 4. Louis, C.A., “A study of groundwater flow in jointed rock and its influence on the stability of rock masses”, Rock Mech. Res. Report, 10 (1969). 5. Snow, D.T., “A parallel plate model of fractured permeable media”, Ph.D. Thesis, Univ. of Calif., Berkeley, U.S.A., p. 331 (1965). 6. Long, R.R., “Mechanics of solids and fluids”, Prentice-Hall, Englewood Cliffs, N. J. (1961). 7. Piggott, A.R., and Elsworth, D., “Analytical models for flow through obstructed domains”, Jour. Geophys. 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