Finite-element simulations of Stoneley guided-wave reflection and scattering at the tips of fluid-filled fractures

Transcription

1 GEOPHYSICS, VOL. 75, NO. 2 MARCH-APRIL 2010 ; P. T23 T36, 12 FIGS., 2 TABLES / Finite-element imulation of Stoneley guided-wave reflection and cattering at the tip of fluid-filled fracture Marcel Frehner 1 and Stefan M. Schmalholz 2 ABSTRACT The reflection and cattering of Stoneley guided wave at the tip of a crack filled with a vicou fluid wa tudied numerically in two dimenion uing the finite-element method. The rock urrounding the crack i fully elatic and the fluid filling the crack i elatic in it bulk deformation behavior and vicou in it hear deformation behavior. The crack geometry, epecially the crack tip, i reolved in detail by the untructured finite-element meh. At the tip of the crack, the Stoneley guided wave i reflected. The amplitude ratio between reflected and incident Stoneley guided wave i calculated from numerical imulation, which provide value ranging between 43% and cloe to 100% depending on the type of fluid filling the crack water, oil or hydrocarbon ga, the crack geometry elliptical or rectangular, and the preence of a mall ga cap at the cracktip. The interference of incident and reflected Stoneley guided wave lead to a node zero amplitude at the tip of the crack. At other poition along the crack, thi interference increae the amplitude. However, the exponential decay away from the crack make the Stoneley guided wave difficult to detect at a relatively hort ditance away from the crack. The part of the Stoneley guided wave that i not reflected i cattered at the crack tip and emitted into the urrounding elatic rock a body wave. For fully aturated crack, the radiation pattern of thee elatic body wave point in every direction from the crack tip. The emitted elatic body wave can allow the detection of Stoneley guided wave-related reonant ignal at ditance away from the crack where the amplitude of the Stoneley guided wave itelf i too mall to be detected. INTRODUCTION Fracture in rock are of great practical interet not only becaue they contribute ignificantly to the permeability of a rock e.g., Faoro et al., 2009 but alo becaue they can have a ignificant influence on eimic wave that pa through fractured rock. For example, Saenger and Shapiro 2002 how with numerical imulation that the wave velocity of body wave decreae dratically with increaing crack denity, Groenenboom and Falk 2000 model numerically and meaure in the laboratory that cattering of body wave at hydraulic fracture i trong enough to determine the fracture dimenion, and Kotek et al and Ionov 2007 how that fracture interecting a borehole can have a major impact in eimic urvey. One phenomenon of particular interet are Stoneley guided wave SGW, a highly diperive and lowly propagating wave mode that i bound to a crack e.g., Ferrazzini and Aki, 1987; Ahour, 2000; Korneev, SGW alo are referred to a crack wave Chouet, 1986; Yamamoto and Kawakatu, 2008, low Stoneley wave Ferrazzini and Aki, 1987 or imply Stoneley wave in a fracture Ahour, They are of interet becaue of their ability to develop a reonance when propagating back and forth along a crack, which hould lead to trongly frequency dependent propagation effect for eimic wave Korneev, Depite their potential importance for wave propagation in porou and fractured rock, SGW are not conidered in exiting effective medium and poroelatic theorie, uch a the Hudon model Hudon, 1980, 1981, the quirt-flow model Mavko and Jizba, 1991; Dvorkin et al., 1995 or the Biot model Biot, Analytical tudie of SGW propagation are available only for infinite traight crack Ferrazzini and Aki, 1987; Ahour, 2000; Korneev, 2008 not taking into account the reflection and cattering at crack tip and therefore alo not taking into account the reonant behavior of SGW. Numerical tud- Preented at the 78thAnnual International Meeting, SEG. Manucript received by the Editor 16 February 2009; revied manucript received 26 June 2009; publihed online 23 March Formerly ETH Zurich, Department of Earth Science, Geological Intitute, Zurich, Switzerland; preently Univerity of Vienna, Department for Geodynamic and Sedimentology, Vienna, Autria. marcel.frehner@univie.ac.at. 2 Formerly ETH Zurich, Geological Intitute, Department of Earth Science, Zurich, Switzerland; preently Univerity of Lauanne, Intitute of Geology and Palaeontology, Lauanne, Switzerland. tefan.chmalholz@unil.ch Society of Exploration Geophyicit. All right reerved. T23

2 T24 Frehner and Schmalholz ie are rare e.g., Chouet, 1986; Yamamoto and Kawakatu, 2008 and available only for imple crack geometrie uually rectangular. Thi paper extend thi body of knowledge by tudying the propagation, reflection, and cattering of SGW at crack tip of different hape and with a high numerical reolution. Becaue SGW are bound to a crack, they are reflected at the crack tip and can propagate back and forth along a crack. The reulting reonance caued by SGW propagating in finite fracture i ued by Akietal. 1977, Chouet 1988, and Chouet 1996 to explain longperiod volcanic tremor ignal that are oberved before volcanic eruption and potentially can be ued for eruption forecating. The reflection coefficient at the crack tip together with the attenuation determine how many time an SGW can propagate back and forth along a crack and, therefore, how well it can develop a reonance. Knowing that SGW cannot be detected at a relatively hort ditance away from the crack due to the exponential decay of their amplitude Ferrazzini and Aki, 1987, the way the tremor ignal i tranmitted to recording tation at the earth urface remained unclear. Ferrazzini andaki 1987 upect that reflection at the crack tip hould provide an important ource of radiation in the cae of a finite crack. However, the reflection of SGW at the crack tip and the correponding radiation of body wave from the crack tip have not been invetigated in detail until now, which i why they are the main ubject of thi paper. The part of the SGW that i not reflected i cattered at the crack tip and P- and S-wave are radiated away from the crack tip. The radiation pattern of thee P- and S-wave i of great importance for meauring the reonant behavior of the SGW i.e., the tremor ignal. Similar to volcanic area, SGW can be of great importance in fractured reervoir. Frequency-dependent wave-propagation phenomena in exploration eimology can help to characterize uburface fractured reervoir. The tudy of SGW i a multicale problem where typical wavelength can be order of magnitude larger than the characteritic ize of the crack. For numerical imulation, thi preent a major computational challenge Korneev, The tandard numerical method for imulating wave propagation in fractured media i the finite-difference method FDM uing a rectangular numerical grid Chouet, 1986; Kotek et al., 1998; Groenenboom and Falk, 2000; Saenger and Shapiro, 2002; Krüger et al., The numerical method ued in thi tudy i the finite-element method FEM e.g., Zienkiewicz and Taylor, 2000; Cohen, 2002 uing an untructured numerical meh. A imilar method alo uing an untructured meh i, for example, the dicontinuou Galerkin method decribed by Käer and Dumber The untructured meh allow reolving geometrically complex object with trong material contrat e.g., the tip of a crack very finely and accurately without the need to have a high reolution elewhere in the domain Frehner et al., In contrat, rectangular grid alway approximate object in a taircae-like way, which lead to numerical inaccuracie no matter how fine the numerical grid i. For time integration in wave-propagation imulation, explicit cheme are mot common. The larget explicit time increment allowed for table numerical olution i determined by the mallet patial reolution and the larget wave velocity in the domain Virieux, 1986; Higham, 1996; Saenger et al., Both parameter take extreme value when SGW are imulated. Spatial reolution need to be very fine around the crack tip and the diperive P-wave in the vicou fluid have a velocity tending to infinity for very high frequencie. Small-amplitude numerical error or noie, which i commonly characterized by high frequencie, can grow during the imulation and lead to numerical intabilitie. One poibility to avoid thee intabilitie in vicou fluid i defining frequency-dependent material parameter Saenger et al., 2005, which allow defining the high-frequency limit of the diperive wave from infinity down to a finite value. The alternative ued in thi tudy i an implicit time-integration method e.g., Chen et al., 2008; Frehner et al., 2008, which doe not require fulfilling any tability criterion. Material parameter in the numerical algorithm can be implemented exactly the ame way they are written in the contitutive equation and do not have to be made frequency dependent. The paper begin with a decription of the mathematical and geometrical model. Propertie of the SGW a a function of the model etup and the different fluid ued in thi tudy are decribed uing analytical expreion of Ferrazzini and Aki 1987 and Korneev A brief introduction to the applied two-dimenional 2D FEM i given before the numerical reult are hown. The reflection of an SGW at the tip of a crack i quantified a a function of crack geometry and the type of fluid filling the crack. The radiation pattern of elatic body wave that are emitted into the urrounding rock i decribed in detail. The paper end with imulation for two advanced model etup two interecting fracture and fracture filled with two different fluid and a dicuion about applicability of the modeling reult to natural environment. MODEL The propagation of SGW i tudied with a 2D model with Carteian coordinate x and y. The mathematical decription and the geometrical etup are decribed below. Mathematical model The 2D formulation ued here i a plane-train approximation of the full 3D formulation, i.e., all patial derivative in the third dimenion, the out-of-plane diplacement, and all out-of-plane train are equal to zero. However, the out-of-plane normal tre i allowed to have a finite value, which depend only on the two in-plane normal train. The plane-train approximation in 2D i equivalent to a 3D formulation with geometrie extending to infinity in the third dimenion i.e., all material parameter are contant in the third dimenion. For thi formulation, tandard material parameter can be ued that are defined in 3D e.g., elatic bulk modulu K. A full 3D formulation i computationally expenive and i not ued in thi tudy. Reult obtained with the 2D plane-train formulation are applicable to crack with a relatively round crack urface i.e., penny-haped crack but not to trongly elongated crack i.e., cigar-haped crack. The force-balance equation or conervation of linear momentum that decribe the tate of the acting force in 2D Love, 1927; Linday, 1960; Achenbach, 1973; Shame and Cozzarelli, 1997; Aki and Richard, 2002; Pujol, 2003 i given by 0 xx u x x y σ!!" ρ = σ yy, u!!" y 0 xy y x σ u!!" #$$%$$& T σ B where i denity and ũ i the econd time derivative of the diplacement vector. The ymbol denote the continuou nature of ũ not yet dicretized with any numerical method. Vector contain the 1

3 Stoneley guided-wave reflection & cattering T25 three independent component ij of the ymmetric total tre tenor i.e., xx, yy, and xy. Compreive tree are defined a negative. Supercript T denote the tranpoe of a matrix. The deformation behavior of the medium i divided into a bulk or volumetric part and a deviatoric or hear part. Therefore, the vector i alo divided into a bulk and a deviatoric part Shame and Cozzarelli, 1997 : σ xx p xx σ yy = p + yy. 2 xy 0 σ xy Vector contain the three independent component ij of the ymmetric deviatoric tre tenor and p i preure or mean tre. Vector, containing the three independent component i.e., two normal component xx and yy, and one hear component xy of the ymmetric total train tenor, i divided into a bulk and a deviatoric part in a imilar way: ε xx ux x Θ 3 e xx ε yy = uy y = Θ 3 + eyy. 3 xy ux y uy x 0 g γ xy ε e Vector e contain the three independent component i.e., two normal component e xx and e yy, and one hear component g xy of the ymmetric deviatoric train tenor and i the bulk train i.e., xx yy zz, where zz i equal to zero due to the plain-train formulation. Two different type of media are conidered in thi tudy: the rock olid, upercript and the fluid upercript f that fill the crack. The behavior of both media i the ame a in Korneev The bulk deformation behavior of both media i linear elatic, while the deviatoric deformation behavior of the two media i different. The deviatoric deformation of the olid rock i linear elatic and that of the fluid i linear vicou. The contitutive equation for the elatic bulk deformation of both media i p K,f, where K,f i the elatic bulk modulu of the olid and the fluid, repectively. The contitutive equation for the deviatoric deformation of the elatic olid i xx yy xy e yy g xy, 0 0 exx where i the elatic hear modulu. The contitutive equation for the vicou deviatoric deformation of the fluid i 4 5 xx 2η 0 0 e! xx yy = e η! yy, 6 xy 0 0 g η! xy e! where i the hear vicoity. Vector ė i the time derivative of vector e. The formulation for total tre in the elatic olid i found by combining equation 2 5 a xx yy xy K 4 /3 K 2 /3 0 K 2 /3 K 4 / ũ x / x ũ y / y ũ x / y ũ y / x. The formulation for total tre in the fluid i found by combining equation 2 4 and 6 a xx yy xy Kf Kf 0 K f K f ũ x / x ũ y / y ũ x / y ũ y / x 4 /3 2 /3 0 2 /3 4 / ũ x/ x 7 ũ y/ y ũ x/ y ũ y/ x. Equation 7 decribe the tre-train relation of a fully elatic medium in 2D where both the bulk and hear deformation behavior are elatic. Therefore, P- and S-wave can propagate in uch a medium without velocity diperion or attenuation. Equation 8 decribe the tre-train relation of a o-called vicoacoutic medium a vicou fluid in 2D. Only the bulk deformation behavior i elatic, while the hear deformation behavior i vicou. Therefore, hear wave exit excluively due to vicoity and have a diffuive character. On the other hand, P-wave can propagate in uch a medium but they are attenuated by the vicou damping term. The formulation i very imilar to the one-dimenional formulation of a medium uing a Kelvin- Voigt model Bourbie et al., 1987; Carcione, The P-wave phae velocity in the fluid V Pf i diperive with a low-frequency limit equal to V C Kf / f. For increaing frequency, the phae velocity increae continuouly and tend to infinity without having a highfrequency limit. The quality factor for P-wave in uch a vicoacoutic fluid Q Pf i equal to infinity no attenuation in the low-frequency limit and Q Pf 0 purely diffuive propagation type in the high-frequency limit. Setting the hear vicoity to 0 lead to a purely acoutic formulation, alo called an invicid fluid. P-wave in an invicid fluid propagate with the velocity V C. They are neither diperive nor attenuated. Equation 7 and 8 can be written in a more general way a D el D vic, 9 where, in the purely elatic cae, D el i the matrix given in equation 7 and D vic i equal to 0. In the vicoacoutic cae, D el i the firt matrix given in equation 8 and D vic i the econd matrix given in equation 8. Equation 9 i ubtituted into equation 1 to yield the total equation of motion: Geometrical model,f ũ B T D el Bũ B T D vic Bũ To imulate SGW and their behavior at a crack tip, the three model etup ketched in Figure 1 are ued. The firt model labeled 1

4 T26 Frehner and Schmalholz conit of a traight horizontal crack of thickne h 3 mm that run through the whole model domain and i centered at y 0. Thi model doe not contain a crack tip. The SGW propagate unditurbed along the crack and can be compared with the analytical olution for the phae velocity Ferrazzini and Aki, 1987; Korneev, The econd model labeled 2 conit of half a crack that ha an elliptical hape with a horizontal major emiaxi L 0.5 m and a vertical minor axi h 3 mm. The tip of the crack i located at x 0 and y 0. The third model labeled 3 conit of a traight horizontal crack of length L 0.5 m and thickne h 3 mm, ending at a flat crack tip i.e., rectangular crack geometry. The tip of the crack i located at x 0 and y 0. In both the econd and the third model, the SGW propagate along the crack and i partly reflected at the crack tip. In all three model etup, two vertical line with virtual receiver recording the diplacement field are located at x/h 70.0 line 1 and x/h 3.3 line 2, repectively. Becaue all model etup are ymmetric around y 0, receiver are only poitioned in the poitive y-direction. In all three model, the boundarie are far enough away from the crack to avoid boundary effect. Rigid wall boundary condition all diplacement u 0 are applied all around the model except for the poition where the crack i in contact with the left boundary. There, only the diplacement in the y-direction i forced to vanih and the diplacement in the x-direction i precribed by the time- and pacedependent boundary condition 2 t t 0 F t,y A 0 2 exp t t for h/2 y h/2 and x L, 1 2 y 2 h 11 which act a the external driving force. Equation 11 implie that the ource initiating the SGW i located inide the crack. The time-dependent part of F t,y i the firt derivative of a Gauian, centered at time t 0. The pace-dependent part of F t,y i a hyperbola with maximum amplitude 1 at y 0 and zero amplitude at y h/2. The applied parameter are A m, , and t Thi external ource contain all frequencie with a central frequency f Hz. Thi central frequency may eem high in the context of exploration eimology. But, on one hand, the central frequency ued here i not meant to be realitic for a ource ued in exploration eimology. Rather, it reflect the fat opening or propagation of a crack. On the other hand, the reonance frequency of an SGW propagating back and forth along a finite crack depend on the wave velocity while the length of the crack, and the central frequency i a function of the applied ource. The central frequency of the ource wa deliberately defined relatively high to be ditinguihably different from the poible reonance frequency of the SGW. MODEL PROPERTIES Figure 1. Sketche of the three model etup labeled 1 to 3 ued for 2D numerical imulation: A crack filled with a fluid i urrounded by an elatic rock. The ketche are not to cale and the apect ratio of the figure i not correct for viualization reaon, i.e., the crack thickne appear much larger than it actually i. All length are normalized with the crack thickne h. Model 1 dahed line i of a traight horizontal crack with contant thickne h 3 mm that run through the whole model domain. Model 2 olid line repreent half a crack that ha an elliptical hape and end inide the model domain. Model 3 tippled line repreent a rectangular crack with contant thickne h 3 mm that end inide the model domain at a flat crack tip. In all model, two virtual vertical receiver line are placed in the poitive y-direction. Hatched wall repreent the rigidwall boundary condition applied all around the model except for the poition where the crack i in contact with the left boundary. There, time-dependent boundary condition, indicated with three arrow, act a the external ource. For the econd and third model etup Figure 1, the apect ratio of the crack i 2L /h 333. In thi tudy, the elatic rock alway ha the ame propertie and different fluid are defined to fill the crack. Table 1 lit the material parameter of the individual media. Propertie for the elatic rock and for water, oil, and hydrocarbon ga agree with value of Ferrazzini and Aki 1987, Mavko et al. 2003, and Korneev Table 2 lit the propertie of the whole model i.e., fluid-filled crack and urrounding rock and of the SGW. For comparion, the two dimenionle parameter C crack tiffne and F vicou damping lo defined by Chouet 1988 a and C Kf 2L h F 12 2L f h 2 V P 12 13

5 Stoneley guided-wave reflection & cattering T27 are given in Table 2. Formula for calculating phae velocitie of the SGW in Table 2 are given by Ferrazzini andaki 1987 and Korneev 2008 for invicid and vicou fluid, repectively, and are not repeated here. For the choen material parameter, according to Korneev 2008, the model etup lie in the thick fracture regime ee Equation 38 of Korneev, Alo, for the etup with an elliptical crack geometry i.e., variable crack thickne, at leat 99% of the crack lie in the thick fracture regime. Only very cloe to the tip, the crack become more narrow and eventually lie in the thin fracture regime. To normalize the different wavelength in the ytem, crack thickne h can be ued. For example, the applied material and ource parameter lead to a ratio of P0 /h 170, where P0 i the wavelength of a P-wave propagating in the elatic olid with the central frequency of the external ource. Thi how that the crack i two order of magnitude thinner than the wavelength of a P-wave in the urrounding rock. The phae velocity of the SGW i a function of the elatic and vicoacoutic parameter of the rock and the fluid filling the crack, a well a the crack thickne and frequency. Diperion curve for both invicid and vicou fluid Ferrazzini and Aki, Table 1. Elatic and vicoacoutic material parameter for the different media ued in thi tudy. Supercript and f denote propertie of the elatic rock (olid) and of the fluid, repectively. The diperive P- and S-wave phae velocity for vicou fluid f f f V P and V S and the correponding quality factor Q P i calculated for the central frequency of the external ource. The f f low-frequency limit of V P (called V C ) i equal to V P for an invicid fluid. The S-wave in the vicou fluid exit excluively due to f vicoity and ha a diffuive propagation type. The correponding quality factor Q S i equal to zero. The P- and S-wave phae velocity for the elatic rock i not diperive and the correponding quality factor Q P and Q S are infinitely large. Medium Solid rock upercript Water Oil Ga Bulk modulu K GPa Ratio K/K Shear modulu GPa 6 Shear vicoity Pa Ratio / water Denity kg/m Ratio / P-wave phae velocity V P m/ Ratio V P /V P Low-frequency limit of V Pf V C m/ Ratio V C /V P f Quality factor of P-wave in vicou fluid Q P S-wave phae velocity V S m/ Ratio V S /V P Table 2. Propertie of the crack and of the SGW for different fluid that fill the crack. Supercript and f denote propertie of the elatic rock (olid) and of the fluid, repectively. All propertie are calculated for the particular value for the crack geometry, the material propertie of the olid and the central frequency of the external ource ued in thi tudy. The phae velocity of the SGW i calculated uing the olution of Ferrazzini and Aki (1987) and Korneev (2008) for invicid and vicou fluid, repectively. Type of fluid Water Oil Ga Crack tiffne C 2K f L / h Vicou damping lo F 12 2L / f h 2 V P Phae velocity of SGW for vicou fluid V SGW m/ Ratio V SGW /V P f Ratio V SGW /V P invicid Phae velocity of SGW for invicid fluid V SGW m/ invicid Ratio V SGW /V P invicid Ratio V SGW /V C Quality factor of SGW Q SGW

6 T28 Frehner and Schmalholz 1987; Korneev, 2008 how a decreae of the phae velocity for low frequencie. For zero frequency, the phae velocity i zero. The highfrequency limit of the phae velocity of the SGW i that of a Scholte wave Carcione and Helle, 2004, which i the interface wave at a ingle fluid-olid interface. Figure 2 how the phae velocity and Figure 3 the quality factor of the SGW propagating along a traight crack firt model in Figure 1 for a range of material parameter of the fluid, together with the parameter ued in thi tudy Table 1 for contant material parameter of the olid and contant value of h and f 0. Analytical formula for producing Figure 2 and 3 are given by Ferrazzini andaki 1987 and Korneev 2008 for invicid and vicou fluid, repectively, and are not repeated here. For material parameter of water, oil and hydrocarbon ga, the abolute phae velocity V SGW for invicid fluid Figure 2a lie within a very narrow range of 0.2 to 0.27 time of the P-wave phae velocity in the elatic olid V P. However, compared to the acoutic P-wave phae velocity in the invicid fluid V C, the phae velocity of the SGW varie coniderably for the different fluid from 0.4 V C for water to 0.98 V C for hydrocarbon ga. Plotting the fluid parameter f veru K f for water, oil, and hydrocarbon ga reult approximately in a traight line in double logarithmic repreentation gray line in Figure 2a. Thi traight line i ued a the abcia in Figure 2b and Figure 3 where the ordinate i the normalized vicoity of the fluid. The vicoitie of water, oil, and hydrocarbon ga are too mall to have a ignificant effect on the phae velocity of the SGW compared to the invicid cae bottom of Figure 2b.At the ame time, the quality factor of the SGW Figure 3 i relatively large more than 100 for the applied fluid vicoitie and only very little attenuation of the SGW i expected. NUMERICAL METHOD The algorithm ued for numerical imulation i an extended verion of the algorithm preented and benchmarked in Frehner et al It employ the finite-element method FEM Hughe, 1987; Bathe, 1996; Zienkiewicz and Taylor, 2000 for dicretization of the patial derivative in equation 10. The particular finite element ued i a even-node ioparametric triangular element with biquadratic continuou interpolation function Zienkiewicz and Taylor, The untructured numerical meh i generated by the oftware Triangle Shewchuk, 1996; Shewchuk, It i generated in uch a way that interface between different media coincide with element boundarie of the finite-element meh. Figure 4 how two ubfigure on different cale of the ame finite-element meh that dicretize the model etup with the elliptical crack. The finite-element algorithm ued comprie the Galerkin weighted-reidual method Zienkiewicz and Taylor, 2000, lumped ma matrix Bathe, 1996; Cohen, 2002, and Gau-Legendre quadrature on even integration point Zienkiewicz and Taylor, Equation 10, dicretized in pace with the FEM, take the form M L ü Cu Ku 0, 14 where M L, C, and K are the lumped ma matrix, the damping matrix, and the tiffne matrix, repectively. Diplacement vector u contain the unknown diplacement u x and u y at all dicrete poition in the finite-element meh. Note that the ymbol ha been removed from u compared to equation 10 becaue it i now dicretized in pace i.e., u contain only the value at numerical node. Time derivative are dicretized with an implicit verion of the Newmark a) b) Figure 2. a Contour line of the SGW phae velocity V SGW for a range of acoutic material parameter of the fluid and for zero vicoity invicid fluid. b Contour line of the SGW phae velocity V SGW for a range of vicoacoutic parameter of the fluid. The abcia of b i a linear relationhip between log 10 f and log 10 K f that approximately connect the material parameter f and K f of water, oil, and hydrocarbon ga and i hown in a a a gray line. In both a and b, V SGW i divided by the P-wave phae velocity in the elatic rock V P and by the P-wave phae velocity in the vicoacoutic fluid V Pf V C for the invicid cae in a. Material parameter for the olid, the crack thickne h, and the central frequency of the wave f 0 are contant in both a and b. Material parameter of the invicid acoutic and vicoacoutic fluid ued in thi tudy water, oil, and hydrocarbon ga are indicated a full and open circle, repectively.

7 Stoneley guided-wave reflection & cattering T29 algorithm Zienkiewicz and Taylor, It i a predictor-corrector algorithm baed on a finite-difference formulation: ü prediction k 1 1 t 2u k 1 t u k ü k, Predictor: u prediction k 1 t u k 1 u k 1 k ; 2 tü 15 Solution: u k 1 C K 1 t 2M L 1 t Corrector: Cu prediction k 1 M L ü prediction k 1 ; ü k 1 ü prediction k 1 1 t 2u k 1, u k 1 u prediction k 1 t u k 1; Subcript k i the index of any dicrete time interval and t i the time increment. For the two Newmark parameter and, the optimal value of 1 4 and 1 2 are choen Newmark, 1959; Bathe, Becaue the time integration method i implicit, no tability criterion for the time increment ha to be fulfilled and the time increment t can be choen independently from the patial reolution. Thi allow having a very fine patial reolution Figure 4 without the need of a very mall time increment. The time increment i choen in uch a way that a P-wave in the elatic rock travel the ditance 2L in 2000 time tep. Spatial reolution i choen in uch a way that the wavelength of the SGW wave i reolved with at leat 80 numerical point. Propagation velocity of the very low diffuion-type S-wave in the vicou fluid i.e., diffuion velocity i order of magnitude lower than all other wave in the model ee Table 1. Therefore, for the imulated time pan, the S-wave in the fluid i quai-tationary. However, the mot important effect of vicoity, i.e., the damping of all the different propagating wave, i correctly imulated in the model. The numerical algorithm i written in MATLAB and the ytem of equation i olved with a tandard direct olver provided by MATLAB. Simulation were performed on one CPU on a tandard worktation. The FEM for wave propagation alo can be ued in combination with explicit finite-difference time-integration cheme or with finite-element time-integration cheme. Different cheme are preented and compared in Frehner et al Benchmark of the numerical code Amodified verion of the numerical code i benchmarked in Frehner et al for a different geometrical etup compriing fully elatic and acoutic media but no vicoacoutic media. Figure 5 how the phae-velocity diperion curve of an SGW calculated for a traight crack and for the model parameter diplayed in the figure. Analytical olution are taken from Ferrazzini and Aki 1987 and Korneev 2008 for acoutic invicid and vicoacoutic fluid, repectively. Five numerical imulation were performed with different central frequencie of the external ource. The model conit- Figure 3. Contour line of the logarithm of the quality factor of the SGW Q SGW for a range of vicoacoutic parameter of the fluid. The abcia i a linear relationhip between log 10 f and log 10 K f that approximately connect the material parameter f and K f of water, oil, and hydrocarbon ga and i hown in Figure 2a a a gray line. Material parameter for the olid, the crack thickne h, and the central frequency of the wave f 0 are kept contant. Material parameter of the vicoacoutic fluid ued in thi tudy water, oil, and hydrocarbon ga are indicated a open circle. Figure 4. Two ubfigure on different cale of the ame numerical finite-element meh dicretizing the model with the elliptical crack econd model in Figure 1. Black element have material parameter of the vicoacoutic fluid. Gray element have material parameter of the elatic rock. The patial reolution of the meh varie trongly, being very fine inide and cloe to the crack. In both ubfigure, it i not poible to how the entire crack and the entire numerical meh becaue the numerical domain i much larger.

8 T30 Frehner and Schmalholz ing of a traight crack Figure 1, firt model i ued for comparion with the analytical olution. The velocity of the SGW calculated from the time hift between meaurement at the two receiver line Figure 1 i plotted on top of the analytical olution. Thee numerically calculated velocitie agree well with the analytical olution. Becaue the ret of the tudy conider only the thick fracture regime defined by Korneev 2008, the benchmark i performed only for thi regime. NUMERICAL RESULTS In the following, reult of the propagation, reflection, and cattering of SGW are preented that are derived from different numerical imulation. Figure 5. Phae velocity diperion curve for an SGW propagating along a traight crack for the diplayed model parameter. Ferrazzini andaki 1987 provide an exact their equation 14b and an approximate their equation 16, alo preented in equation 1 in Korneev, 2008 olution for an infinite crack filled with an invicid acoutic fluid. In hi equation 40, Korneev 2008 provide a olution for a crack filled with a vicoacoutic fluid. Numerical reult are derived from five imulation uing the firt model in Figure 1 with different central frequencie f 0 in the external ource. The phae velocity of the SGW V SGW i normalized with the phae velocity of a P-wave in the rock V P. The frequency f i normalized with V P /h. Radiation of elatic body wave from the crack tip The SGW i bound to the crack and cannot propagate further when the crack end. It mut be partly reflected at the crack tip. Figure 6 how the naphot of a imulation of an SGW propagating from left to right along an elliptical crack econd model in Figure 1 filled with vicou water. Panel a and b how the incident SGW, which i almot unaffected by the preence of the crack tip. Becaue the crack thin toward the crack tip due to it elliptical hape, the SGW low down toward the crack tip. Therefore, even though the a) c) e) g) b) d) f) h) Figure 6. Snaphot of the 2D diplacement field of a imulation of an SGW propagating along an elliptical crack econd model in Figure 1 filled with vicou water. Contour line are defined at value m, m, m, m, and m. Upper panel how the x-component of the diplacement field u x. Lower panel how the y-component of the diplacement field u y. Panel from left to right repreent progreive point in time, with time indicated between the panel. Axi label are only given in the left and lower panel but are valid for all panel. The SGW i partially reflected at the crack tip and elatic body wave are emitted from the crack tip into the urrounding rock.

9 Stoneley guided-wave reflection & cattering T31 SGW ha not reached the crack tip yet, it i lightly deformed at it front. The regular pacing of the logarithmically plotted contour line demontrate the exponential decay of the amplitude away from the crack Ferrazzini and Aki, The amplitude decay more than one order of magnitude within one wavelength of the SGW. Panel c and d how the SGW a it tart being reflected at the crack tip.alo, a part of the wave energy i tranferred to the urrounding elatic rock in the form of body wave. The part of the body wave in panel c propagating parallel to the crack away from the crack tip i.e., along the line y 0 i a P-wave becaue the diplacement direction and the propagation direction are parallel. All other viible body wave propagating with a certain angle to the crack away from the crack tip are combination of P- and S-wave. The exact geometry of the P- and S-wave i not calculated from the diplacement field and i not diplayed here. In panel e and f, the incident and reflected wavetrain of the SGW interfere detructively and the amplitude cloe to the crack tip i relatively mall. Panel g and h how the final phae of the reflection proce. The SGW now propagate from right to left away from the crack tip. Interetingly, the radiation pattern of the body wave around the crack tip point in every direction from the crack tip, which lead to the interpretation that the crack tip act like a point diffractor for the SGW. Thi interpretation can be undertood becaue the width of the crack and, therefore, the ize of the crack tip, are order of magnitude maller than the wavelength of the SGW. In all panel of Figure 6, the interference of the incoming and reflected SGW train lead to a node zero amplitude exactly at the crack tip. Therefore, the reflection pattern of the SGW can be compared to a reflection of a 1D wave propagating in a medium with lower impedance at the interface to a medium with higher impedance. Figure 7 how naphot of a imulation of an SGW propagating from left to right along a traight crack with a flat crack tip third model in Figure 1 filled with vicou water. Unlike in Figure 6, the phae velocity of the SGW doe not change along the crack due to the contant thickne of the crack. Therefore, the individual naphot in Figure 7 are not diplayed for the ame point in time a in Figure 6 but it wa tried to diplay the ame tage of the reflection proce to make Figure 6 and 7 comparable. The reflection pattern of the SGW at the flat crack tip i very imilar to the one at the elliptical crack tip. However, the wavetrain i not compreed toward the crack tip becaue the SGW doe not low down toward the crack tip. Similar to the elliptical crack tip, the radiation pattern of body wave around the flat crack tip point in every direction from the crack tip at the end of the reflection proce. However, a major difference between the two geometrical etup i the amplitude of thee body wave in the elatic olid, with the amplitude being coniderably higher for a flat crack tip. Reflection of the SGW at the crack tip A een above, not all of the wave energy of the SGW i reflected at the crack tip but a part i radiated into the urrounding rock in the form of elatic body wave. Figure 8 diplay the diplacement-time ignal at two receiver on receiver line 1 Figure 1, one inide and one outide the crack, for a imulation of an SGW being reflected at a) c) e) g) b) d) f) h) Figure 7. Same a Figure 6 but for a rectangular crack with a flat crack tip third model in Figure 1. Note that the time of the naphot i not the ame a in Figure 6 becaue the SGW travel with a lightly different velocity.

10 T32 Frehner and Schmalholz the tip of an elliptical crack econd model in Figure 1 filled with vicou water. The incident and reflected SGW are well eparated from each other in time. To quantify the reflected part of the SGW, Figure 9 how the amplitude ratio R between reflected and incident SGW for different model etup and for different fluid filling the crack. Here, the term reflection coefficient i avoided becaue not only the amplitude change but alo the hape of the SGW when it i reflected. Value of R are calculated from the diplacement-time ignal at receiver on receiver line 1 Figure 1, like the example hown in Figure 8. For each imulation two value for R are calculated, one at receiver inide the crack in the vicoacoutic fluid and one for receiver outide the crack in the elatic rock. Value labeled oil with ga cap, elliptical crack tip are dicued later. Value plotted for material propertie of water value to the right of Figure 9 correpond to the two imulation hown in Figure 6 and 7. The amplitude of the SGW reflected at the tip of an elliptical crack filled with water i around 77% of the amplitude of the incident SGW and only around 43% when reflected at the flat crack tip. Thi i remarkable becaue the ize of the crack tip i order of magnitude maller than the wavelength of the SGW but till ha a big impact. The difference in reflection behavior alo explain the amplitude difference of the radiated body wave hown in Figure 6 and 7. The part that i not reflected i radiated into the urrounding rock. Therefore, a tronger reflection elliptical crack lead to maller amplitude of the radiated body wave. For different fluid filling the elliptical crack, the reflection i alo different. Hydrocarbon ga lead to the tronget reflection with R of almot 100%. Thi alo mean that from a crack filled with hydrocarbon ga, only mall-amplitude body wave are radiated when the SGW i reflected at the crack tip. An SGW propagate both in the fluid that fill the crack and in the rock urrounding the crack. It i therefore unclear how to calculate the impedance for an SGW. However, the trong reflection for a crack filled with hydrocarbon ga can be qualitatively undertood by conidering the impedance of the P-wave in the fluid Kf f, which i much maller for hydrocarbon ga than for water and oil. Therefore, the impedance contrat to the urrounding rock i much bigger, which lead to a trong reflection. a) Due to the interference between incident and reflected SGW, the amplitude add up cloe to the crack tip. Figure 10 how thi effect and how the amplitude decay away from the crack along receiver line 2 ee Figure 1 for different model etup and different fluid filling the crack. The amplitude ditribution how the ame exponential decay a dicued in Ferrazzini and Aki A a reference olid gray line, the amplitude decay along receiver line 1 for an elliptical crack filled with water i alo given in Figure 10. For thi cae, the wavelength of the SGW i around 40 time the crack thickne h. At thi ditance away from the crack, the amplitude decay i more than an order of magnitude. The amplitude at the crack interface at receiver line 2 for a water or oil-filled crack i increaed by about 30% due to the interference between incident and reflected SGW. Alo, the two crack geometrie elliptical and rectangular crack that are filled with water do not influence thi factor ignificantly. The ame amplitude for a crack filled with hydrocarbon ga i increaed by about 120%. Thi i remarkable becaue for a reflection a trong a R 100% Figure 9, a maximal increae in amplitude cloe to the crack tip of 100% i expected. However, the wave velocity of the SGW alo decreae toward the crack tip due to the elliptical hape of the crack. Thi let the amplitude of the SGW further increae, which add up to the maximal 100% increae in amplitude due to the reflection proce. For all cae hown in Figure 10, even though the amplitude at the crack interface i increaed, the exponential decay away from the crack happen within a relatively hort Vicou fluid Elliptical crack tip Oil with ga cap Elliptical crack tip Flat crack tip b) Figure 8. a Diplacement-time ignal in the x-direction at a receiver inide the crack on receiver line 1 Figure 1. b Diplacementtime ignal in the y-direction at a receiver outide the crack on receiver line 1. Both trace are obtained from a imulation of an SGW propagating along an elliptical crack econd model in Figure 1 filled with vicou water. Label for the time axi are only given in b but are valid for a alo. Figure 9. Abolute value of the amplitude ratio R between reflected and incident SGW of an SGW that i reflected at the tip of a crack. The abcia i the ame a in Figure 2b and Figure 3. R x i calculated from the diplacement-time ignal in the x-direction of eight receiver inide the crack on receiver line 1 at poition y/h 0 to R y i calculated from the diplacement-time ignal in the y-direction of ix receiver outide the crack on receiver line 1 at poition y/h 0.45 to 5.5. Value labeled vicou fluid, elliptical crack tip are derived from imulation of an elliptical crack econd model in Figure 1 fully aturated with the correponding vicou fluid. Value labeled flat crack tip are derived from a imulation of a rectangular crack with a flat crack tip third model in Figure 1 fully aturated with vicou water. Value labeled oil with ga cap, elliptical crack tip are derived from a imulation of an elliptical crack econd model in Figure 1 partially aturated with vicou oil and having a mall ga cap at the crack tip. Thee value are plotted at K Oil K Ga /2. All value of R are corrected for the intrinic attenuation due to vicou damping in the fluid.

11 Stoneley guided-wave reflection & cattering T33 ditance. For crack filled with water or oil, the amplitude along receiver line 2 i even maller than along receiver line 1 for ditance greater than five time the crack thickne. Advanced model etup The model etup coniting of an elliptical crack econd model in Figure 1 i ued for imulating a partially filled crack. The crack i filled with vicou oil and ha a mall cap at the crack tip filled with hydrocarbon ga. The ga cap extend from x/h 31.8 to x/h 0. Figure 11 how the naphot of the diplacement field in the x- and y-direction after the SGW i reflected at the crack tip.amajor part of the SGW i reflected already at the oil-ga contact line and only a mall-amplitude SGW propagate further along the crack where it i reflected at the crack tip. Thi multiple reflection lead to the complex reflection pattern in Figure 11. One major difference to the crack filled only with oil almot identical to the crack filled only with water, Figure 6 i the amplitude and radiation pattern of the elatic body wave that are radiated away from the crack tip when the SGW i reflected. The radiation pattern i much more forwarddirected toward the propagation direction of the incident SGW, compared to a radiation pattern pointing in every direction for the fully aturated crack Figure 6. Alo, the amplitude of the radiated body wave are much larger. Figure 9 how the amplitude ratio R between reflected and incident SGW for both cae. For the crack fully aturated with oil, R i about 78%. It i reduced to about 43% when the ga cap i preent. The larger amplitude of the radiated body wave alo mean that le of the energy of the SGW i reflected compared to the fully aturated crack. Ditributed individual and iolated crack are only one poible crack pattern in nature. More common are probably warm of imilarly oriented crack or two or more familie of crack whoe orientation interect. Figure 12 how two naphot at different point in time of a imulation of two interecting crack. The firt crack, in which the SGW i initiated, ha an apect ratio of 333. The econd crack ha an apect ratio of 95. The angle between the two crack i 60. The firt naphot Figure 12a i taken before the SGW reache the interection point of the two crack. Two SGW train propagated away from the external ource in the crack. The left wavetrain i already reflected at the left crack tip and now both wavetrain are propagating toward the interection point to the right. Alo viible are the elatic body wave that propagate in the urrounding rock away from the external ource and are cattered by the crack. The econd naphot Figure 12b i taken after the firt SGW train paed the interection point of the two crack. Only a part of the SGW continue propagating traight ahead along the firt crack. A part i reflected at the interection point and interfere with the econd SGW train on the firt crack. A coniderable part of the SGW make a harp turn and propagate along the two branche of the econd crack. Alo, elatic body wave are radiated away from the interection point into the urrounding rock. DISCUSSION Model of SGW propagating along fluid-filled crack on variou cale are ued to explain the occurrence of long-period volcanic tremor Aki et al., 1977; Chouet, 1988; Chouet, The magma chamber a a whole or fracture around the volcanic conduit can be conidered a the waveguide where an SGW propagate back and forth, which reult in a characteritic frequency. Becaue the SGW amplitude decay exponentially away from the crack, the way thi a) Colorcode for diplacement field in [m] Water, elliptical, line 1 Water, elliptical, line 2 Oil, elliptical, line 2 Ga, elliptical, line 2 Water, flat, line 2 b) Figure 10. Maximum abolute particle diplacement along receiver line 1 and 2 Figure 1 recorded during four different imulation. Three imulation are for an elliptical crack filled with three different vicou fluid. The fourth imulation i for a rectangular crack ending at a flat crack tip filled with vicou water. Maximum abolute particle diplacement along receiver line 1 i only hown for the elliptical crack filled with water olid gray line becaue it i almot identical for all imulation. All value of one imulation are normalized with the maximum abolute particle diplacement at the crack interface at receiver line 1. Figure 11. a Snaphot of the x- and b y-component of the 2D diplacement field of a imulation of an SGW propagating along an elliptical crack econd model in Figure 1. The crack i filled with vicou oil and ha a mall ga cap at the crack tip. Contour line are the ame a in Figure 6 and 7.Axi label for the abcia are only given in b but are valid in a alo.

12 T34 Frehner and Schmalholz long-period ignal i tranmitted to recording tation at the earth urface remained unclear. The emiion of elatic body wave dicued in thi tudy make it poible to detect SGW-related ignal even in ditance away from the crack where the amplitude of the SGW itelf i too mall to be meaured. Which type of body wave i.e., P- or S-wave i more important remain to be determined in a future tudy. The orientation of fully aturated crack or magma chamber might not be determinable from meaurement of volcanic tremor due to the radiation pattern that point in every direction from the crack tip, but it might be poible for crack containing a ga cap. Depending on the type of magma, vicoitie can vary by order of magnitude, but in general, they are coniderably larger than that of the fluid conidered in thi tudy. Depending on the magma vicoity and the crack thickne, the quality factor of an SGW lie between 1 and 100. The reflection of an SGW at the tip of a crack can till be trong. However, an SGW in a thin crack filled with a highly vicou magma i expected to be attenuated relatively fat and cannot propagate back and forth along the crack everal time. Conequently, no long-period volcanic tremor will be generated. If longperiod volcanic tremor are a reult of SGW falling into reonance, a) b) Figure 12. a and b Snaphot of the 2D diplacement field at two different point in time of a imulation of an SGW propagating along an elliptical crack that i interected by a econd elliptical crack. The diplayed value i the normalized abolute particle diplacement 10 6 ux 2 u y 2. The crack are filled with vicou water.axi label for the abcia are only given in b but are valid in a alo. it i more likely to oberve them when lower-vicoity magma are preent in thicker crack. Still, it i unlikely that a ingle SGW would propagate back and forth along the crack many time and produce a continuou long-period volcanic tremor. For thi, a continuou excitation of SGW would be neceary. Reervoir rock for hydrocarbon often contain a large number of fracture. The network of fracture contribute ignificantly to the permeability of a reervoir. Kotek et al and Ionov 2007 demontrate that fracture can have an important effect in borehole eimology. Alo, SGW-related effect uch a rock-internal reonance can be important for monitoring hydrofracturing procee during the exploration of hydrocarbon reervoir. However, current model for poroelatic and fractured rock do not include thee effect. Thi and future tudie will help to include SGW-related effect into more realitic model for fractured rock Korneev et al., Frehner et al develop a baic model that couple reonance effect with eimic wave propagation. Thi model wa applied to ocillation that can take place on the pore level due to urface-tenion effect in partially aturated porou rock. However, the reonant behavior of SGW in fractured rock i another poible explanation for the wave propagation-ocillation model preented in Frehner et al Model need to be deigned for a whole finite crack where both crack tip are fully reolved to imulate the propagation of SGW back and forth along the crack and the development of the correponding reonance frequency. Korneev 2009 how that ocillation in the uburface can be meaured with a eimic array. Ocillation are eaier to detect in late arrival when they are not maked by high-energy body wave. Thi implie that long-lating ocillation are more eaily detectable than hort-lating ocillation. In the cae of an SGW, thi mean that a trong reflection at the crack tip eventually enable the detection of the reonant character of the SGW. All preented imulation ue a ource inide the crack. Poible caue of a ource inide the crack are, for example, the opening or propagation of the crack due to magma migration in a volcanic area Chouet, 1986 or hydrofracturing of a uburface reervoir that i under production. For a ource inide the crack, it i clear that an SGW i initiated. It remain to be determined whether a ource outide the crack, e.g., a plane P- or S-wave in the elatic urrounding rock, i capable of initiating an SGW with ignificant amplitude. Thi important next tep will help to undertand how body wave are influenced by SGW in fractured rock. Becaue SGW can generate reonance in finite crack, it can be expected that there are frequency-dependent effect on body wave, e.g., attenuation and diperion. Undertanding how trong thee effect are i eential for cae where body wave propagate through fluid-aturated fractured rock, uch a in exploration eimology or ite effect analyi of earthquake data. Epecially in exploration eimology, a better undertanding of the reonating SGW in a fractured reervoir can help to determine fracture-related petrophyical parameter uch a fracture length, fracture orientation, or fluid vicoity. The preented numerical model deal with a multicale wavepropagation phenomenon with length cale of different order of magnitude. Although highly reolved, the numerical etup i till rather imple, coniting of only one ingle crack. In thi tudy, one approach toward more realitic model etup i hown by modeling two highly reolved interecting crack. Another approach i, for example, a model of many crack Saenger and Shapiro, 2002; Saenger et al., The primary invetigation target of uch model i to determine effective bulk rock propertie. However, the high patial