ResearchArticle PP- and Split PS-Wave AVA Responses of ...wave). e polarization directions of the split PS- and PS-waves are parallel and vertical to the fracture plane, respectively.IntheHTImodel,theelasticpropertiesofthe

$Page 1: ResearchArticle PP- and Split PS-Wave AVA Responses of ...wave). e polarization directions of the split PS- and PS-waves are parallel and vertical to the fracture plane, respectively.IntheHTImodel,theelasticpropertiesofthe$
Research ArticlePP- and Split PS-Wave AVA Responses of Fractured Shale

Shuai Yang and Jun Lu

Key Laboratory of Marine Reservoir Evolution and Hydrocarbon Accumulation Mechanism Ministry of EducationChina University of Geosciences Beijing 100083 China

Correspondence should be addressed to Jun Lu lujun615163com

Received 18 December 2017 Revised 24 March 2018 Accepted 19 April 2018 Published 17 May 2018

Academic Editor Xiao-Qiao He

Copyright copy 2018 Shuai Yang and Jun Lu This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

With seismic exploration advancing to the deep Earth the seismic response of fractured strata has currently become a hot researchtopic However characteristics of the amplitude variation with angle (AVA) of saturated fractured shale still remain unclearFurthermore the direct relationships between the AVA response and fracture system parameters have not received much attentionThis study is aimed at analyzing the effect of fracture density on AVA responses of fractured shale For this purpose we proposea method for modeling saturated fractured shale and analyzing AVA responses of PP- and split PS-waves First we introduceGurevichrsquos fluid theory into the fractured-shale modeling and establish the relationship between Thomsenrsquos weak anisotropyparameters and fracture density and fluid properties Second we perform forward simulation considering an isotropic overburdenand a fractured stratum The results show that differences in AVA responses of the fractured-shale model and the isotropic modelincrease with increasing fracture densities At small to intermediate incidence angles the reflection coefficients of split PS-wavesincrease whereas those of PP-wave decrease The reflection coefficients of the two models differ dramatically at incidence angleslarger than 55∘ Furthermore when the fracture density is large polarity reversal occurs only in PS2-wave AVA gathers

1 Introduction

With the recent intensification of exploration and develop-ment of unconventional oil and gas reservoirs geophysicalprospecting methods of shale reservoirs have been receivingmuch attention worldwide Among such methods the AVO(amplitude variationwith offset) orAVA (amplitude variationwith angle) technique is considered as one of the mostpromising techniques as it has played an important rolein conventional oil and gas exploration Castagna et al [1]studied AVO intercepts and gradients for pairs of shalegassand Avseth et al [2] derived the shale trend for a gas-and-oilfield of North Sea through rock physics modeling and AVOanalysis However thus far AVO studies of shales have onlyconsidered isotropic media and ignored seismic anisotropyleading to deviations in shale reservoir characterization Infact due to the compaction of the overburden high-anglefractures are widely developed in shale reservoirs whichserve as important reservoir spaces and seepage channels forshale gas Because of tectonic stress high-angle fractures tendto be aligned along the same direction Such reservoir models

can be described as a kind of anisotropic media namedHTI media (transversely isotropic media with a horizontalaxis of symmetry) [3] There are three modes of body wavepropagation in this media a P-wave two shear-waves withdifferent velocities and mutually orthogonal polarizationdirections [4 5] These two shear-waves are split from thesame shear wave when its polarization direction is oblique tothe fracture plane

Some scholars have studied the AVO responses of HTImedia However these studiesmainly focused on the applica-tion of P-wave AVO responses to the prediction of fracturedreservoirs Ruger and Tsvankin [6 7] used the P-waveazimuthal AVO method to detect fractures Ruger [8 9]further extended the P-wave reflection coefficient to HTImedia Perez et al [10] studied the practical application of P-wave AVA to the detection of fracture orientation Gray andHead [11] studied the properties of anisotropic media basedon the P-wave azimuthal anisotropy and predicted fracturesin an actual oilfield Hall and Kendall [12] applied the P-wave AVA of a submarine three-component seismic datasetto analyze natural fracture characteristics Ruger [13 14]

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 1587634 9 pageshttpsdoiorg10115520181587634

2 Mathematical Problems in Engineering

PS2

PS1

isotropy

plane

symmetryaxisP-wave

PP

Figure 1 Schematic of the HTI model

proposed the approximate reflection coefficient equations forHTI media usingThomsenrsquos anisotropy parameters

Previous AVO researches on fractured reservoirs aremainly based on commonly used fracture anisotropymodelsHudsonmodel [15] and linear slip model [16] However bothmodels assume that the background medium embedded inthe fracture is a single elastic solid medium without consid-ering pores and fluids In reality the presence of pore fluidsand wave-induced fluid flow between pores and fracturesstrongly influence the propagation of elastic waves Recentstudies have proposed an HTI model that considers theproperties of both fluid flowand seismic anisotropyThomsen[17 18] proposed the theories of weak seismic anisotropyand used five parameters to characterize reservoir anisotropyThomsen considered fractured rocks as an isotropic modelcontaining a series of fractures connected with the same porediameter and fluids flow between the fractures and poresThis theory is more applicable for computing the seismic-wave characteristics of the fluid-filled fracture model in theseismic scale Chapman [19] proposed the fracture modelswith both pore and fracture dimensions Cardona [20] usedthe anisotropic Gassmann equations to achieve a fluid inde-pendent replacement of the geometrical shape of fracturesGurevich [21] further developed the formula of the fluidsubstitution interpretation for the vertical fractured mediawith the fluids in pores Gurevichrsquos formula can establish theexact analytical relationships between the effective stiffnessmatrix of a rock and the elastic properties of the dry back-ground porosity fracture compliances and the saturatingfluid However studies on the direct relationship betweenseismic-wave responses and fracture system parameters suchas fracture density and fillings are lacking

In this study based on Gurevichrsquos theory of saturatingfluid we established the relationship between Thomsenrsquosanisotropy and elastic parameters and fracture systemparam-eters Both fluidity and the anisotropy were considered in theproposed HTI model Then we built a theoretical fractured-shale model and applied the reflection coefficient equationsderived by Ruger to analyze PP- and split PS-wave AVAresponses

2 Methodology

21MathematicalModel There are four common anisotropicmedia in the Earth [22] Triclinic medium has no symme-try planes or an axis of cylindrical symmetry monoclinicmedium has only a symmetry plane orthorhombic mediumhas three mutually orthogonal planes of symmetry trans-versely isotropicmediumhas an axis of cylindrical symmetryTransversely isotropic medium is the simplest anisotropicmedium which is referred to as the HTI medium whenthe symmetry axis is horizontal The anisotropy of an HTImedium in the oil and gas exploration is always induced bya single fracture set embedded in an isotropic matrix [14]Therefore we used the HTI model to establish single-layershale with aligned vertical fractures as the model illustratedin Figure 1 The fine layering model with vertical fractureswas not considered because it involves a more complexanisotropic model with multigroup and multiscale fractures

In a multicomponent seismic survey P-wave sources areused to initiate seismic waves As shown in Figure 1 thereflected P-wave (PP-wave) propagates through the fracturezone without changing the original polarization directionWhen the original polarization direction is oblique to the

Mathematical Problems in Engineering 3

fracture (isotropy) plane the reflected S-wave (PS-wave)splits into a fast mode (PS1-wave) and a slow mode (PS2-wave) The polarization directions of the split PS1- andPS2-waves are parallel and vertical to the fracture planerespectively In the HTI model the elastic properties of thefractured shale can be described using the following stiffnesscoefficient matrix [14]

C =[[[[[[[[[[[[

11988811 11988813 11988813 0 0 011988813 11988833 (11988833 minus 211988844) 0 0 011988813 (11988833 minus 211988844) 11988833 0 0 00 0 0 11988844 0 00 0 0 0 11988855 00 0 0 0 0 11988855

]]]]]]]]]]]] (1)

Elastic and anisotropy parameters related to AVA responsescan be converted by the stiffness coefficients

120572 = radic 11988833120588 120573 = radic 11988844120588

120576(119881) = 11988811 minus 11988833211988833 120575(119881) = (11988813 + 11988855)2 minus (11988833 minus 11988855)2211988833 (11988833 minus 11988855) 120574(119881) = 11988855 minus 11988844211988844 120574 = minus2120574(119881) + 1120574(119881)

(2)

where the elastic parameters 120572 120573 and 120588 denote the isotropy-plane velocities of the vertical P- and S1-waves and densityrespectively 120576(119881) and 120575(119881) are the anisotropy parameterscharacterizing the P-wave velocity anisotropy 120574(119881) is theanisotropy parameter indicating the fractional differencebetween the velocities of S2- and S1-waves The superscriptldquo119881rdquo denotes vertical direction along the fracture plane

Regarding saturated fractured shale individual fluid sub-stitution expressions for each stiffness coefficient of the HTImodel are as follows [21 23]

119888sat11 = 119871119863 [1198891120578 + 119870119891120593119870119898119871 (11987111198871015840 minus 1612058321198870Δ1198739119871 )] 119888sat33 = 119871119863 [1198892120578 + 119870119891120593119870119898119871 (11987111198871015840 minus 412058321198870Δ1198739119871 )] 119888sat13 = 120582119863 [1198891120578 + 119870119891120593119870119898120582 (12058211198871015840 minus 812058321198870Δ1198739119871 )] 119888sat44 = 120583119888sat55 = 120583 (1 minus Δ119879)

(3)

where

119871 = 120582 + 2120583119863 = 1 + 119870119891119870119898120593 (1198870 minus 120593 + 1198702Δ119873119870119898119871 ) 120578 = 1 minus 119870119891119870119898 120593 = 120593119898 (1 minus 120593119888) + 120593119888120593119888 = 41205871198861198903 1198870 = 1 minus 119870119870119898 1198871015840 = 1198870 + 1198702Δ119873119870119898119871 1198711 = 119870119898 + 431205831205821 = 119870119898 minus 231205831198891 = 1 minus Δ1198731198892 = 1 minus 1205822Δ1198731198712 119870 = 120582 + 21205833

(4)

and 119870 is the bulk modulus of background material 119870119891is the fluid bulk modulus which can be calculated by thebulk moduli of water (119870119908) gas (119870119892) and oil (119870119900) with thecorresponding saturations 119878119908 119878119892 and 119878119900

1119870119891 = 119878119908119870119908 + 119878119900119870119900 + 119878119892119870119892 (5)

In (4) 119871 119863 120578 1205821 1198891 1198892 1198870 and 1198871015840 are the intermediatevariables 120582 and 120583 are the Lame parameter and the shearmodulus of the background isotropic material respectively120593 is the total porosity of the fractured rock and 120593119898 and120593119888 are the porosities of background material and fracturesrespectively 119886 is the aspect ratio of the fracture 119890 is thefracture density 119870119898 is the solid grain bulk modulus Δ119873and Δ119879 are the normal and tangential fracture weaknesseswhich can be calculated by the dimensionless normal (119864119873)and tangential (119864119879) compliances of the fracture systemrespectively


Δ119879 = 1198641198791 + 119864119879 Δ119873 = 1198641198731 + 119864119873 119864119873 = 41198903119892 [1 minus 119892 + (1198961015840 + 412058310158403) (120587119886120583)] 119864119879 = 161198903 (3 minus 2119892 + 41205831015840120587119886120583) 119892 = 120583120582 + 2120583

(6)

where 1198961015840 and 1205831015840 are the bulk and shear moduli of the fillingsin the fractures respectively

Equations (3)ndash(6) provide a complete description of thestiffness coefficients of the saturated fractured HTI modelThen the elastic and anisotropy parameters can be expressedin terms of the properties of dry (isotropy) backgroundfillings in fractures and saturating fluid by substitutingsaturated rock stiffness coefficient equations (3)ndash(6) into (2)

120572 = radic91198711198634 minus 4119861Δ1198739119863119871120588 120573 = radic120583120588 120576(119881) = 12119861Δ119873 + 9 (1198632 minus 1198631) 11987128119861 minus 181198711198634 120575(119881)= 3 [minus3 (1198634 + 1198633) 119871 + 4119861Δ119873] [4119861Δ119873 + 9119871 (1198634 minus 1198633 minus 2119863119879)]2 (minus91198634119871 + 4119861Δ119873) [4119861Δ119873 minus 9119871 (1198634 minus 119863119879)] 120574 = Δ1198792 (1 minus Δ119879)

(7)

where

1198631 = (1 minus Δ119873) (1 minus 119870119891119870119898) 1198632 = (1 minus 12058221198712Δ119873)(1 minus 119870119891119870119898) 1198633 = 119870119891120593119870119898 (119870119898 minus 23120583)(1198870 + 1198702Δ119873119870119898119871 )

+ 120582 (1 minus Δ119873) (1 minus 119870119891119870119898) 1198634 = 119870119891120593119870119898 (119870119898 + 43120583)(1198870 + 1198702Δ119873119870119898119871 )

+ (1 minus 12058221198712Δ119873)(1 minus 119870119891119870119898) (120582 + 2120583)

119861 = 11988701199062 119870119891120593119870119898 119879 = 120583 (1 minus Δ119879) 120588 = (1 minus 120593) 120588119898 + 120593 (119878119908120588119908 + 119878119900120588119900 + 119878119892120588119892) (8)

and 119861 119879 1198631 1198632 1198633 and 1198634 are the intermediate variablesThen we can obtain the elastic and anisotropy parametersfor establishing the saturated HTI model if the backgroundmedium parameters of 120582 120583119870119898 120593119898 and 120588119898 (the backgroundmedium density) fracture system parameters of 119886 119890 1198961015840 and1205831015840 and saturating fluid parameters of 119870119908 119870119892 119870119900 119878119908 119878119892 119878119900120588119908 120588119892 and 120588119900 are known22 Reflection Coefficients To study the AVA response char-acteristics of the saturated HTI model the reflection coeffi-cients of PP- and split PS-waves can be calculated by Rugerrsquosequations [14]

119877PP = 12 Δ119885119885 + 12 Δ120572120572 minus (2120573120572 )2 Δ119866119866

+ [[(120575(119881)2 minus 120575(119881)1 ) + 2(2120573120572 )2 (1205742 minus 1205741)]] cos2 120579sdot sin2 1198941 + 12 Δ120572120572 + (1205762(119881) minus 1205761(119881)) cos4 120579+ (120575(119881)2 minus 120575(119881)1 ) sin2 120579 cos2 120579 sin2 1198941tan2 1198941

119877PS1 = minus12 Δ120588120588 sin 1198941cos 1198951 minus 120573120572 (Δ120588120588 + 2Δ120573120573 ) sin 1198941 cos 1198941

+ (120573120572)2 (2Δ120573120573 + Δ120588120588 ) sin3 1198941

cos 1198951 119877PS2 = minus12 (Δ120588120588 ) sin 1198941

cos 1198951 minus 120573120572 Δ120588120588+ 2(Δ120573120573 + 1205741 minus 1205742) sin 1198941 cos 1198941 + (120573120572)

2

sdot 2(Δ120573120573 + 1205741 minus 1205742) + Δ120588120588 sin3 1198941cos 1198951

+ [[[(12057222 (1205722 minus 1205732) cos 1198951 minus

120572120573 cos 11989412 (1205722 minus 1205732))sdot (1205752(119881) minus 1205751(119881))]]] sin 1198941


+ [[[120572120573 cos 1198941(1205722 minus 1205732) (1205752(119881) minus 1205751(119881) + 1205761(119881) minus 1205762(119881))]]]

sdot sin3 1198941 minus [[[(1205722(1205722 minus 1205732) cos 1198951)

sdot (1205752(119881) minus 1205751(119881) + 1205761(119881) minus 1205762(119881))]]] sin3 1198941+ [[[(

12057322 (1205722 minus 1205732) cos 1198951)(1205751(119881) minus 1205752(119881))]]] sin3 1198941+ [[[(

1205732(1205722 minus 1205732) cos 1198951)

sdot (1205752(119881) minus 1205751(119881) + 1205761(119881) minus 1205762(119881))]]] sin5 1198941(9)

where

120572 = 12 (1205721 + 1205722) Δ120572 = 1205722 minus 1205721120573 = 12 (1205731 + 1205732)

Δ120573 = 1205732 minus 1205731120588 = 12 (1205881 + 1205882)

Δ120588 = 1205882 minus 1205881119885 = 120588120572119866 = 1205881205732

(10)

Here 1198941 is the incidence phase angle of the P-wave 1198951 isthe emergence phase angle of the S1-wave 1198941 and 1198951 followSnellrsquos Law [14] 120579 is the azimuth angle of the P-wave Thesubscripts ldquo1rdquo and ldquo2rdquo denote parameters of the upper andlower strata respectively Equations (9)-(10) relate the elasticand anisotropy parameters with the PP- and split PS-wavereflection coefficients

3 Model Analysis

The goal of this study was to analyze the effect of fracturedensity on the AVA responses of the fractured-shale modelWe performed forward simulations based on the theoreticalmodel consisting of an isotropic overburden and a fracturedstratum considering the same parameters for the backgroundoverburden (Table 1) For the fractured stratum the physicalquantities of the fracture system and the saturating fluidsare shown in Table 1 Following the procedure shown inFigure 2 wemodeled fractured shale (Table 2) and calculatedthe reflection coefficients of the PP- and split PS-waves(Figure 3) Here fluids considered in the fractured shale onlyinclude gas and water As shown in Table 2 with fracturedensity increasing from 0 to 04 the vertical P- and S-wavevelocities decreased by about 45 and 28 respectivelydensity decreased by about 86 120576(119881) and 120575(119881) had negativevalues and 120574 was positive which are physically reasonable[14]

31 AVA Analysis As shown in Figure 3 the curves ofvariations in the reflection coefficients of the PP- or PS2-waves are sharper than that of the PS1-wave Figure 3(a)shows that the reflection coefficients of PP-wave increasewithincreasing incidence angle and then decrease significantly atincidence angles larger than 50∘ Furthermore the fluctuationof the PP-wave reflection coefficient curve is more obvious athigh fracture density This is because in the lower fracturedstratum 120572 120573 120588 120576(119881) and 120575(119881) decrease and 120574 increases asfracture density increases Moreover differences in elasticand anisotropy parameters between the upper and lowerstrata increase Figures 3(b) and 3(c) show that the variationtrends of the reflection coefficients of the PS1- and PS2-wavesare consistent They increase first and then decrease withincreasing incidence angle In addition variation rate of thereflection coefficient increases at higher fracture densitiesThere are two intersections between the reflection coefficientcurves of the PS1- and PS2-waves with one common inter-section at the incidence angle of 0∘ For the PS1-wave theother intersection is near the incidence angle of 65∘ and forthe PS2-wave the other intersection is at the incidence angleof 55∘ The reflection coefficients of the PS2-wave vary frompositive to negative values but those of the PS1-wave remainpositive

We compared AVA responses of the fractured-shalemodel with those of shale without fractures The shale with-out fractures can be considered as a type of isotropic mediummodel for which anisotropy parameters are zero In additionin an isotropic medium model the shear wave will not besplit For comparison we calculated relative differences in thereflection coefficients of the PP-wave between the fractured-shale and isotropic models (Figure 4(a)) The PS-wave reflec-tion coefficient of the isotropic model was the same as thePS1-wave reflection coefficient of the fractured-shale modelTherefore only the calculated relative differences between thePS-wave reflection coefficients of the isotropic model and thePS2-wave reflection coefficients of the fractured-shale modelare shown in Figure 4(b)


Calculate the elastic and anisotropy parameters accordingto equations (6) (7) and (8)

AVA response analysis

Generate reflection coefficientsof PP- PS1- PS2-waves

Build the saturated fractured shale models

Generate the AVA gathers ofPP- PS1- PS2-waves

Input the parameters of saturated fluid and backgroundmaterials and fractures Km m m a k KwKg Ko Sw Sg So w g Ｈ＞ o

Calculate DKf Ｈ＞ c according to equations (4) and (5)

Figure 2 Flowchart of the fractured-shale modeling and AVA response analysis

Table 1 Parameters of the background medium fractures and fluids

Fluid parameters Values Background medium parameters Values Fracture parameters Values119870119908 (GPa) 238 119870119898 (GPa) 1908ndash1768 1198961015840 (GPa) 918GPa119870119892 (GPa) 002 120588119898 (gcm3) 251 1205831015840 (GPa) 603GPa120588119908 (gcm3) 109 120583 (GPa) 1138ndash1054 119890 001ndash040120588119892 (gcm3) 014 120582 1138ndash1054 119886 001119878119908 50 120593119898 10119878119892 50

Table 2 Different fracture density models of shale

Fracture density 120572 (kms) 120573 (kms) 120588 (gcm3) 120576(119881) 120575(119881) 1205740 385 221 25 0 0 0001 3839 2217 2311 minus0003 minus0005 0003010 3798 2201 2304 minus0030 minus0050 0029020 3756 2183 2297 minus0055 minus0090 0057040 3679 2149 2283 minus0097 minus0152 0111

As shown in Figure 4(a) with increasing incidence anglethe absolute value of the relative differences initially increasethen decrease and dramatically increase towards the endAlthough the velocities and densities of the two modelsare the same changes in anisotropic parameters will greatlyaffect the reflection coefficient In Figure 4(b) the relativedifference between the two models becomes more apparentat higher fracture density

In the fractured-shale model within incidence anglesof 40∘ higher fracture density can lead to large changes inthe PS2-wave reflection coefficients but small changes in thePP-wave reflection coefficients However at incidence angles

over 55∘ the reflection coefficients of both the PP- and PS-waves change dramatically

32 AVA Gathers Analysis According to the model parame-ters in Table 2 AVA gathers were synthesized by convolutionusing the Ricker wavelet with a dominant frequency of 40Hzand the reflection coefficients shown in Figure 3 AVA gathersof the PP-wave with different fracture densities are shown inFigures 5(a) 5(b) 5(c) and 5(d) The reflection amplitudesare enhanced with increasing fracture density especially atlarge incidence angles over 60∘ For small fracture density asshown in Figure 5(a) the reflection amplitudes decrease with


001 010

020 040

Fracture density

80 0 30 50 60 70 40 10 20 Incidence angle (degrees)

minus020

minus016

minus012

minus008

minus004

000 R００

(a)

001 010

020 040

Fracture density


000

004

008

012

016

020

R０３1

(b)

001 010

020 040

Fracture density


minus008

minus004

000

004

008

R０３2

(c)

Figure 3 AVA response curves of the PP-wave (a) PS1-wave (b) and PS2-wave (c) as a function of incidence angles and fracture densities

001 010

020 040

Fracture density

minus05

00

05

10

15

20

Relat

ive d

iffer

ence

ofR

００


(a)

001 010

020 040

Fracture density


minus20

minus15

minus10

minus05

00

05

10

Relat

ive d

iffer

ence

ofR

０３

(b)

Figure 4 Relative differences in the reflection coefficients of the PP-wave (a) and PS-wave (b) between the fractured-shale model and theisotropic model


(k) PS2-wave e = 02 (l) PS2-wave e = 04

(i) PS2-wave e = 001 (j) PS2-wave e = 01

(g) PS1-wave e = 02 (h) PS1-wave e = 04

(e) PS1-wave e = 001 (f) PS1-wave e = 01

(c) PP-wave e = 02 (d) PP-wave e = 04

(a) PP-wave e = 001 (b) PP-wave e = 01

10 20 30 40 50 60 700 80

Tim

e (m

s)

10 20 30 40 50 60 700 80Incidence angle (degrees) Incidence angle (degrees)

150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0

Figure 5 AVA gathers of PP- PS1- and PS2-waves as a function of different fracture densities

increasing incidence angle As shown in Figures 5(b) 5(c)and 5(d) the PP-wave reflection amplitudes first increase andthen decrease with increasing incidence angle The PS1-waveAVA gathers in Figures 5(e) 5(f) 5(g) and 5(h) clearly showstrong reflection amplitudes at intermediate incidence anglesHowever therewas no significant change inAVAgatherswithvarying fracture densities In Figures 5(i) 5(j) 5(k) and 5(l)the PS2-wave AVA gathers show strong reflection amplitudes

at intermediate incidence angles However polarity reversaloccurs at fracture densities of 02 and 04

4 Conclusions

In this study we determined the relationship between Gure-vichrsquos fluid theory and Thomsenrsquos anisotropy parameters forthe saturated-fractured-shale model and simulated PP- and


split PS-wave AVA responses From analyses of the reflectioncoefficients of the saturated-fractured-shale and isotropicmodels we found that anisotropy parameters increase withincreasing fracture density and the difference in reflectioncoefficients become more prominent The reflection coeffi-cients of the two models were distinct at incidence angleshigher than 55∘ The differences in reflection coefficientsbetween the two models grew wider as fracture densityincreased Moreover the differences in reflection coefficientswere more dramatic for PP- or PS2-waves than for PS1-waves According to the synthesized AVA gathers we foundthat polarity reversal occurs only in PS2-wave AVA gatherswith high fracture density The modeling of water-filled oroil-filled fractures remains a major unresolved issue Thevelocities and densities of water and oil are close and thustheir corresponding influences on elastic and anisotropyparameters are similar Therefore discriminating the AVAresponse anomaly caused by water and oil is still a problemNevertheless the proposed simulations of the fractured-shalemodel will lay the foundation for further studies on thewavefield characteristics of actual fractured reservoirs

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors greatly appreciate the support of the NaturalScience Foundation of China (41574126 and 41425017) andthe Fundamental Research Funds for the Central Univer-sities (2-9-2017-452) They would like to thank Editage(httpswwweditagecom) for English language editing

References

[1] J P Castagna H W Swan and D J Foster ldquoFramework forAVO gradient and intercept interpretationrdquo Geophysics vol 63no 3 pp 948ndash956 1998

[2] P Avseth A Draege A-J van Wijingaarden T A Johansenand A Joslashrstad ldquoShale rock physics and implications for AVOanalysis a North Sea demonstrationrdquoThe Leading Edge vol 27no 6 pp 788ndash797 2008

[3] LThomsen ldquoElastic anisotropy due to aligned cracks in porousrockrdquo Geophysical Prospecting vol 43 no 6 pp 805ndash829 1995

[4] S Crampin ldquoSeismic-wave propagation through a crackedsolid polarization as a possible dilatancy diagnosticrdquo TheGeophysical Journal of the Royal Astronomical Society vol 53no 3 pp 467ndash496 1978

[5] S Crampin ldquoEvaluation of anisotropy by shear-wave splittingrdquoGeophysics vol 50 no 1 pp 142ndash152 1985

[6] A Ruger and I Tsvankin ldquoAzimuthal variation of AVOresponse for fractured reservoirs 65th annual internationalmeetingrdquo SEG Expanded Abstracts pp 1103ndash1106 1995

[7] A Ruger and I Tsvankin ldquoUsing AVO for fracture detectionanalytic basis and practical solutionsrdquoThe Leading Edge vol 16no 10 p 1429 1997

[8] A Ruger ldquoP-wave reflection coefficients for transverselyisotropicmodels with vertical and horizontal axis of symmetryrdquoGeophysics vol 62 no 3 pp 713ndash722 1997

[9] A Ruger ldquoVariation of P-wave reflectivity with offset andazimuth in anisotropic mediardquo Geophysics vol 63 no 3 pp935ndash947 1998

[10] M A Perez R L Gibson and M N Toksoz ldquoDetection offracture orientation using azimuthal variation of P-wave AVOresponsesrdquo Geophysics vol 64 no 4 pp 1253ndash1265 1999

[11] D Gray and K Head ldquoFracture detection in Manderson fielda 3-D AVAZ case historyrdquo The Leading Edge vol 19 no 11 pp1214ndash1221 2000

[12] S A Hall and J-M Kendall ldquoFracture characterization atValhall application of P-wave amplitude variation with offsetand azimuth (AVOA) analysis to a 3D ocean-bottom data setrdquoGeophysics vol 68 no 4 pp 1150ndash1160 2017

[13] A Ruger ldquoAnalytic insight into shear-wave AVO for fracturedreservoirs 66th annual international meetingrdquo Society of Explo-ration Geophysicists pp 159ndash185 1996

[14] A Ruger ldquoReflection coefficients and azimuthal AVO analysisin anisotropic mediardquo Society of Exploration Geophysicists pp63ndash181 2002

[15] J A Hudson ldquoA higher order approximation to the wavepropagation constants for fractured solidrdquo Geophysical JournalInternational vol 87 no 1 pp 265ndash274 1986

[16] M Schoenberg ldquoReflection of elastic waves from periodicallystratified media with interfacial sliprdquo Geophysical Prospectingvol 31 no 2 pp 265ndash292 1983

[17] L Thomsen ldquoWeak elastic anisotropyrdquo Geophysics vol 51 no10 pp 1954ndash1966 1986

[18] LThomsen ldquoConverted-wave reflection seismology over inho-mogeneous anisotropic mediardquo Geophysics vol 64 no 3 pp678ndash690 1999

[19] M Chapman ldquoModeling the effect of multiple sets of mesoscalefractures in porous rock on frequency-dependent anisotropyrdquoGeophysics vol 74 no 6 pp 97ndash103 2009

[20] R Cardona ldquoTwo theories for fluid substitution in porous rockswith aligned fractures 72nd annual internationalmeetingrdquo SEGExpanded Abstracts pp 173ndash176 2002

[21] B Gurevich ldquoElastic properties of saturated porous rocks withaligned fracturesrdquo Journal of Applied Geophysics vol 54 no 3-4pp 203ndash218 2003

[22] S Crampin ldquoSuggestions for a consistent terminology forseismic anisotropyrdquo Geophysical Prospecting vol 37 no 7 pp753ndash770 1989

[23] L Brown and B Gurevich ldquoFrequency-dependent seismicanisotropy of porous rocks with penny-shaped cracksrdquo Explo-ration Geophysics vol 35 no 2 pp 111ndash115 2004

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Submit your manuscripts atwwwhindawicom


PS2

PS1

isotropy

plane

symmetryaxisP-wave

PP

Figure 1 Schematic of the HTI model

proposed the approximate reflection coefficient equations forHTI media usingThomsenrsquos anisotropy parameters

Previous AVO researches on fractured reservoirs aremainly based on commonly used fracture anisotropymodelsHudsonmodel [15] and linear slip model [16] However bothmodels assume that the background medium embedded inthe fracture is a single elastic solid medium without consid-ering pores and fluids In reality the presence of pore fluidsand wave-induced fluid flow between pores and fracturesstrongly influence the propagation of elastic waves Recentstudies have proposed an HTI model that considers theproperties of both fluid flowand seismic anisotropyThomsen[17 18] proposed the theories of weak seismic anisotropyand used five parameters to characterize reservoir anisotropyThomsen considered fractured rocks as an isotropic modelcontaining a series of fractures connected with the same porediameter and fluids flow between the fractures and poresThis theory is more applicable for computing the seismic-wave characteristics of the fluid-filled fracture model in theseismic scale Chapman [19] proposed the fracture modelswith both pore and fracture dimensions Cardona [20] usedthe anisotropic Gassmann equations to achieve a fluid inde-pendent replacement of the geometrical shape of fracturesGurevich [21] further developed the formula of the fluidsubstitution interpretation for the vertical fractured mediawith the fluids in pores Gurevichrsquos formula can establish theexact analytical relationships between the effective stiffnessmatrix of a rock and the elastic properties of the dry back-ground porosity fracture compliances and the saturatingfluid However studies on the direct relationship betweenseismic-wave responses and fracture system parameters suchas fracture density and fillings are lacking

In this study based on Gurevichrsquos theory of saturatingfluid we established the relationship between Thomsenrsquosanisotropy and elastic parameters and fracture systemparam-eters Both fluidity and the anisotropy were considered in theproposed HTI model Then we built a theoretical fractured-shale model and applied the reflection coefficient equationsderived by Ruger to analyze PP- and split PS-wave AVAresponses

2 Methodology

21MathematicalModel There are four common anisotropicmedia in the Earth [22] Triclinic medium has no symme-try planes or an axis of cylindrical symmetry monoclinicmedium has only a symmetry plane orthorhombic mediumhas three mutually orthogonal planes of symmetry trans-versely isotropicmediumhas an axis of cylindrical symmetryTransversely isotropic medium is the simplest anisotropicmedium which is referred to as the HTI medium whenthe symmetry axis is horizontal The anisotropy of an HTImedium in the oil and gas exploration is always induced bya single fracture set embedded in an isotropic matrix [14]Therefore we used the HTI model to establish single-layershale with aligned vertical fractures as the model illustratedin Figure 1 The fine layering model with vertical fractureswas not considered because it involves a more complexanisotropic model with multigroup and multiscale fractures

In a multicomponent seismic survey P-wave sources areused to initiate seismic waves As shown in Figure 1 thereflected P-wave (PP-wave) propagates through the fracturezone without changing the original polarization directionWhen the original polarization direction is oblique to the



C =[[[[[[[[[[[[

11988811 11988813 11988813 0 0 011988813 11988833 (11988833 minus 211988844) 0 0 011988813 (11988833 minus 211988844) 11988833 0 0 00 0 0 11988844 0 00 0 0 0 11988855 00 0 0 0 0 11988855

]]]]]]]]]]]] (1)


120572 = radic 11988833120588 120573 = radic 11988844120588


(2)




(3)

where


(4)


1119870119891 = 119878119908119870119908 + 119878119900119870119900 + 119878119892119870119892 (5)



Δ119879 = 1198641198791 + 119864119879 Δ119873 = 1198641198731 + 119864119873 119864119873 = 41198903119892 [1 minus 119892 + (1198961015840 + 412058310158403) (120587119886120583)] 119864119879 = 161198903 (3 minus 2119892 + 41205831015840120587119886120583) 119892 = 120583120582 + 2120583

(6)




(7)

where


+ 120582 (1 minus Δ119873) (1 minus 119870119891119870119898) 1198634 = 119870119891120593119870119898 (119870119898 + 43120583)(1198870 + 1198702Δ119873119870119898119871 )

+ (1 minus 12058221198712Δ119873)(1 minus 119870119891119870119898) (120582 + 2120583)

119861 = 11988701199062 119870119891120593119870119898 119879 = 120583 (1 minus Δ119879) 120588 = (1 minus 120593) 120588119898 + 120593 (119878119908120588119908 + 119878119900120588119900 + 119878119892120588119892) (8)


119877PP = 12 Δ119885119885 + 12 Δ120572120572 minus (2120573120572 )2 Δ119866119866



+ (120573120572)2 (2Δ120573120573 + Δ120588120588 ) sin3 1198941

cos 1198951 119877PS2 = minus12 (Δ120588120588 ) sin 1198941


2


+ [[[(12057222 (1205722 minus 1205732) cos 1198951 minus







1205732(1205722 minus 1205732) cos 1198951)


where

120572 = 12 (1205721 + 1205722) Δ120572 = 1205722 minus 1205721120573 = 12 (1205731 + 1205732)

Δ120573 = 1205732 minus 1205731120588 = 12 (1205881 + 1205882)

Δ120588 = 1205882 minus 1205881119885 = 120588120572119866 = 1205881205732

(10)


3 Model Analysis






















001 010

020 040

Fracture density


minus020

minus016

minus012

minus008

minus004

000 R００

(a)

001 010

020 040

Fracture density


000

004

008

012

016

020

R０３1

(b)

001 010

020 040

Fracture density


minus008

minus004

000

004

008

R０３2

(c)


001 010

020 040

Fracture density

minus05

00

05

10

15

20

Relat

ive d

iffer

ence

ofR

００


(a)

001 010

020 040

Fracture density


minus20

minus15

minus10

minus05

00

05

10

Relat

ive d

iffer

ence

ofR

０３

(b)









10 20 30 40 50 60 700 80

Tim

e (m

s)


150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0




4 Conclusions






Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018










C =[[[[[[[[[[[[

11988811 11988813 11988813 0 0 011988813 11988833 (11988833 minus 211988844) 0 0 011988813 (11988833 minus 211988844) 11988833 0 0 00 0 0 11988844 0 00 0 0 0 11988855 00 0 0 0 0 11988855

]]]]]]]]]]]] (1)


120572 = radic 11988833120588 120573 = radic 11988844120588


(2)




(3)

where


(4)


1119870119891 = 119878119908119870119908 + 119878119900119870119900 + 119878119892119870119892 (5)



Δ119879 = 1198641198791 + 119864119879 Δ119873 = 1198641198731 + 119864119873 119864119873 = 41198903119892 [1 minus 119892 + (1198961015840 + 412058310158403) (120587119886120583)] 119864119879 = 161198903 (3 minus 2119892 + 41205831015840120587119886120583) 119892 = 120583120582 + 2120583

(6)




(7)

where


+ 120582 (1 minus Δ119873) (1 minus 119870119891119870119898) 1198634 = 119870119891120593119870119898 (119870119898 + 43120583)(1198870 + 1198702Δ119873119870119898119871 )

+ (1 minus 12058221198712Δ119873)(1 minus 119870119891119870119898) (120582 + 2120583)

119861 = 11988701199062 119870119891120593119870119898 119879 = 120583 (1 minus Δ119879) 120588 = (1 minus 120593) 120588119898 + 120593 (119878119908120588119908 + 119878119900120588119900 + 119878119892120588119892) (8)


119877PP = 12 Δ119885119885 + 12 Δ120572120572 minus (2120573120572 )2 Δ119866119866



+ (120573120572)2 (2Δ120573120573 + Δ120588120588 ) sin3 1198941

cos 1198951 119877PS2 = minus12 (Δ120588120588 ) sin 1198941


2


+ [[[(12057222 (1205722 minus 1205732) cos 1198951 minus







1205732(1205722 minus 1205732) cos 1198951)


where

120572 = 12 (1205721 + 1205722) Δ120572 = 1205722 minus 1205721120573 = 12 (1205731 + 1205732)

Δ120573 = 1205732 minus 1205731120588 = 12 (1205881 + 1205882)

Δ120588 = 1205882 minus 1205881119885 = 120588120572119866 = 1205881205732

(10)


3 Model Analysis






















001 010

020 040

Fracture density


minus020

minus016

minus012

minus008

minus004

000 R００

(a)

001 010

020 040

Fracture density


000

004

008

012

016

020

R０３1

(b)

001 010

020 040

Fracture density


minus008

minus004

000

004

008

R０３2

(c)


001 010

020 040

Fracture density

minus05

00

05

10

15

20

Relat

ive d

iffer

ence

ofR

００


(a)

001 010

020 040

Fracture density


minus20

minus15

minus10

minus05

00

05

10

Relat

ive d

iffer

ence

ofR

０３

(b)









10 20 30 40 50 60 700 80

Tim

e (m

s)


150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0




4 Conclusions






Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018









Δ119879 = 1198641198791 + 119864119879 Δ119873 = 1198641198731 + 119864119873 119864119873 = 41198903119892 [1 minus 119892 + (1198961015840 + 412058310158403) (120587119886120583)] 119864119879 = 161198903 (3 minus 2119892 + 41205831015840120587119886120583) 119892 = 120583120582 + 2120583

(6)




(7)

where


+ 120582 (1 minus Δ119873) (1 minus 119870119891119870119898) 1198634 = 119870119891120593119870119898 (119870119898 + 43120583)(1198870 + 1198702Δ119873119870119898119871 )

+ (1 minus 12058221198712Δ119873)(1 minus 119870119891119870119898) (120582 + 2120583)

119861 = 11988701199062 119870119891120593119870119898 119879 = 120583 (1 minus Δ119879) 120588 = (1 minus 120593) 120588119898 + 120593 (119878119908120588119908 + 119878119900120588119900 + 119878119892120588119892) (8)


119877PP = 12 Δ119885119885 + 12 Δ120572120572 minus (2120573120572 )2 Δ119866119866



+ (120573120572)2 (2Δ120573120573 + Δ120588120588 ) sin3 1198941

cos 1198951 119877PS2 = minus12 (Δ120588120588 ) sin 1198941


2


+ [[[(12057222 (1205722 minus 1205732) cos 1198951 minus







1205732(1205722 minus 1205732) cos 1198951)


where

120572 = 12 (1205721 + 1205722) Δ120572 = 1205722 minus 1205721120573 = 12 (1205731 + 1205732)

Δ120573 = 1205732 minus 1205731120588 = 12 (1205881 + 1205882)

Δ120588 = 1205882 minus 1205881119885 = 120588120572119866 = 1205881205732

(10)


3 Model Analysis






















001 010

020 040

Fracture density


minus020

minus016

minus012

minus008

minus004

000 R００

(a)

001 010

020 040

Fracture density


000

004

008

012

016

020

R０３1

(b)

001 010

020 040

Fracture density


minus008

minus004

000

004

008

R０３2

(c)


001 010

020 040

Fracture density

minus05

00

05

10

15

20

Relat

ive d

iffer

ence

ofR

００


(a)

001 010

020 040

Fracture density


minus20

minus15

minus10

minus05

00

05

10

Relat

ive d

iffer

ence

ofR

０３

(b)









10 20 30 40 50 60 700 80

Tim

e (m

s)


150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0




4 Conclusions






Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018













1205732(1205722 minus 1205732) cos 1198951)


where

120572 = 12 (1205721 + 1205722) Δ120572 = 1205722 minus 1205721120573 = 12 (1205731 + 1205732)

Δ120573 = 1205732 minus 1205731120588 = 12 (1205881 + 1205882)

Δ120588 = 1205882 minus 1205881119885 = 120588120572119866 = 1205881205732

(10)


3 Model Analysis






















001 010

020 040

Fracture density


minus020

minus016

minus012

minus008

minus004

000 R００

(a)

001 010

020 040

Fracture density


000

004

008

012

016

020

R０３1

(b)

001 010

020 040

Fracture density


minus008

minus004

000

004

008

R０３2

(c)


001 010

020 040

Fracture density

minus05

00

05

10

15

20

Relat

ive d

iffer

ence

ofR

００


(a)

001 010

020 040

Fracture density


minus20

minus15

minus10

minus05

00

05

10

Relat

ive d

iffer

ence

ofR

０３

(b)









10 20 30 40 50 60 700 80

Tim

e (m

s)


150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0




4 Conclusions






Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018


























001 010

020 040

Fracture density


minus020

minus016

minus012

minus008

minus004

000 R００

(a)

001 010

020 040

Fracture density


000

004

008

012

016

020

R０３1

(b)

001 010

020 040

Fracture density


minus008

minus004

000

004

008

R０３2

(c)


001 010

020 040

Fracture density

minus05

00

05

10

15

20

Relat

ive d

iffer

ence

ofR

００


(a)

001 010

020 040

Fracture density


minus20

minus15

minus10

minus05

00

05

10

Relat

ive d

iffer

ence

ofR

０３

(b)









10 20 30 40 50 60 700 80

Tim

e (m

s)


150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0




4 Conclusions






Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018









001 010

020 040

Fracture density


minus020

minus016

minus012

minus008

minus004

000 R００

(a)

001 010

020 040

Fracture density


000

004

008

012

016

020

R０３1

(b)

001 010

020 040

Fracture density


minus008

minus004

000

004

008

R０３2

(c)


001 010

020 040

Fracture density

minus05

00

05

10

15

20

Relat

ive d

iffer

ence

ofR

００


(a)

001 010

020 040

Fracture density


minus20

minus15

minus10

minus05

00

05

10

Relat

ive d

iffer

ence

ofR

０３

(b)









10 20 30 40 50 60 700 80

Tim

e (m

s)


150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0




4 Conclusions






Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018















10 20 30 40 50 60 700 80

Tim

e (m

s)


150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Tim

e (m

s)

150

100

50

0




4 Conclusions






Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018












Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








Documents

ResearchArticle PP- and Split PS-Wave AVA Responses of ...wave). e polarization directions of the split PS- and PS-waves are parallel and vertical to the fracture plane, respectively.IntheHTImodel,theelasticpropertiesofthe