張煜, 徐義賢, 夏江海, 張雙喜, 平萍6,

1 武漢大學(xué)測繪學(xué)院, 武漢 430079 2 武漢大學(xué)地球空間環(huán)境與大地測量教育部重點實驗室, 武漢 430079 3 武漢大學(xué)地球空間信息科學(xué)協(xié)同創(chuàng)新中心, 武漢 430079 4 中國地質(zhì)大學(xué)(武漢)多尺度地球成像湖北省重點實驗室, 武漢 430074 5 中國地質(zhì)大學(xué)(武漢)地質(zhì)過程與礦產(chǎn)資源國家重點實驗室, 武漢 430074 6 中國科學(xué)院測量與地球物理研究所大地測量與地球動力學(xué)國家重點實驗室, 武漢 430074

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含流體孔隙介質(zhì)中面波的傳播特性及應(yīng)用

張煜1,2,3,4, 徐義賢4,5, 夏江海4, 張雙喜1,2,3, 平萍6,4

1 武漢大學(xué)測繪學(xué)院, 武漢 430079 2 武漢大學(xué)地球空間環(huán)境與大地測量教育部重點實驗室, 武漢 430079 3 武漢大學(xué)地球空間信息科學(xué)協(xié)同創(chuàng)新中心, 武漢 430079 4 中國地質(zhì)大學(xué)(武漢)多尺度地球成像湖北省重點實驗室, 武漢 430074 5 中國地質(zhì)大學(xué)(武漢)地質(zhì)過程與礦產(chǎn)資源國家重點實驗室, 武漢 430074 6 中國科學(xué)院測量與地球物理研究所大地測量與地球動力學(xué)國家重點實驗室, 武漢 430074

基于單相介質(zhì)中地震波理論的高頻面波法已廣泛應(yīng)用于求取淺地表S波的速度.然而水文地質(zhì)條件表明，普遍的淺地表地球介質(zhì)富含孔隙.孔隙中充填的流體會顯著地影響面波在淺地表的傳播，進而造成頻散和衰減的變化.本文研究了地震勘探頻段內(nèi)針對含流體孔隙介質(zhì)邊界條件的面波的傳播特性.孔隙流體在自由表面存在完全疏通、完全閉合以及部分疏通的情況.孔隙單一流體飽和時，任何流體邊界條件下存在R1模式波，與彈性介質(zhì)中的Rayleigh波類似，相速度稍小于S波并在地震記錄中顯示強振幅.由于介質(zhì)的內(nèi)在衰減，R1在均勻半空間中也存在頻散，相速度和衰減在不同流體邊界下存在差異.Biot固流耦合系數(shù)(孔隙流體黏滯度與骨架滲透率之比)控制頻散的特征頻率，高耦合系數(shù)會在地震勘探頻帶內(nèi)明顯消除這種差異.介質(zhì)的迂曲度等其他物性參數(shù)對不同流體邊界下的R1波的影響也有不同的敏感度.完全閉合和部分疏通流體邊界下存在R2模式波，相速度略低于慢P波.在多數(shù)條件下，如慢P波在時頻響應(yīng)中難以觀察到.但是在耦合系數(shù)較低時會顯現(xiàn)，一定條件下甚至?xí)苑俏锢聿ㄐ问浇邮誖1波的輻射，顯示強振幅.淺表風(fēng)化層低速帶存在，震源激發(fā)時的運動會顯著影響面波的傳播.對于接收點徑向運動會造成面波的Doppler頻移，橫向運動會造成面波的時頻畸變.孔隙存在多相流體時，中觀尺度下不均勻斑塊飽和能很好地解釋體波在地震頻帶內(nèi)的衰減.快P波受到斑塊飽和顯著影響，R1波與快P波有更明顯關(guān)聯(lián)，與完全飽和模型中不同，也更易于等效模型建立.頻散特征頻率受孔隙空間不同流體成分比例變化的控制，為面波方法探測淺地表流體分布與遷移提供可能性.通常情況孔隙介質(zhì)頻散特征頻率較高，標(biāo)準(zhǔn)線性黏彈性固體可以在相對低頻的地震勘探頻帶內(nèi)等效表征孔隙介質(zhì)中R1波的傳播特征，特別在時域，可在面波成像反演建模中應(yīng)用.

流體; 孔隙介質(zhì); 面波; 頻散; 衰減

The pore fluid behaviors through the free surface make the boundary more complex as fully drained, fully sealed and partially drained conditions. These effects are taken into account in surface wave secular equation derivation and closed-form dynamic response investigation. Complex searching algorithm and fast coverage quadrature are applied in the dispersion and response calculation. Typical near surface earth media are selected. For weakly consolidated media in which the surface wave propagates in an extremely low velocity, the effects of the motion of source are obtained. The partial saturation of mesoscopic loss significant in seismic frequency band is introduced beyond the Biot model. The effective viscoelastic model that has been used for body waves in porous media is extended to solve the boundary value problem of surface wave propagation by coupling body wave representations on the free surface.

For one fluid saturated, R1 mode wave can propagate under each boundary condition, which is similar with classic Rayleigh wave in elastic media. Its phase velocity is a little less than the S wave and amplitude is strong in seismograms. Due to the intrinsic attenuation, R1 wave is dispersive in the homogeneous half space. The velocities and attenuation coefficients are different under different fluid boundary conditions. Biot solid-fluid coupling coefficient controls the critical frequency of dispersion as for the body waves. High solid-fluid coupling can eliminate the differences. Tortuosity affects R1 waves for different fluid boundaries with different sensitivities. R2 mode appears and propagates under sealed and partially drained conditions. Its phase velocity is a little less than P2 wave. In most cases, it is difficult to be observed in dynamic responses. However, when the solid-fluid coupling is low, it may obtain the radiation from R1 wave as a non-physical wave with strong amplitude under partially drained surface. When the velocity of surface wave is low in weak media, the movement of source impulse with regards to the receiver makes remarkable effects on surface wave responses. Radical velocity to the receiver makes Doppler frequency shift. Lateral velocity makes distortions of wave responses in time and frequency domains. For partial saturation with heterogeneous multiphase fluid saturated patches, the mesoscopic flow induced by wave can interpreter wave attenuation in seismic frequency band, which is accordance with the practical data. Because the fast P1 wave is dominantly affected by this strong loss mechanism, R1 wave is more distinctively related to P1 wave considering partial saturation, quite different to the Biot fluid saturated model. The critical frequency for mesoscopic mechanism is controlled by fluid compositions in the patches, which in turn presents the correlation for surface wave signals to subsurface fluid distribution and flowing. Because the Biot critical frequency is always high, standard linear viscoelastic solid can effectively represent R1 wave propagation in porous media in seismic frequency band by the coupling body wave fits and ignoring the slow wave perturbation, especially for the dynamic responses in the time domain. While in patchy saturation model, the P wave propagates under the interactive wave induced flow between the two P wave modes, which also make it simple to effectively represent.

The fluid phase in near surface media can be regarded as an important factor for surface wave data acquisition and processing. The effects of different fluid free surfaces need to be considered for correction in several unconsolidated and high permeability rocks. The low velocity of the surface wave makes the movement of source be an important impact in the frequency responses. For passive surface wave observation usually with low frequency signal extraction, this impact needs to be carefully estimated. The dominated mesoscopic mechanism in seismic frequency band shows obvious attenuation in the P wave, which also dominates the surface wave propagation. For near surface attenuation estimation by surface wave, the P wave must be paid attention to. The developed effective medium saves the storage and time consumption for surface wave in porous media, which can be applied in the inversion modeling for surface wave image in an economic way.Keywords Fluid; Porous media; Surface wave; Dispersion; Attenuation

1 引言

體波(P波和S波)在地表干涉產(chǎn)生沿自由表面?zhèn)鞑サ拿娌?Rayleigh波、Love波等).通過反演頻散的面波速度，可獲得不同尺度地下S波速度(Aki and Richards, 1980).面波方法已廣泛地應(yīng)用于水文、工程、環(huán)境、地質(zhì)和地球物理等科學(xué)研究中.但該方法普遍基于彈性介質(zhì)中面波的傳播特性，經(jīng)典線彈性理論是面波資料采集、處理和反演的理論基礎(chǔ)(Song et al., 1989; Misiek et al., 1997; Xia et al., 1999, 2002, 2003, 2004, 2006a, 2006b, 2007, 2008, 2009; Winsborrow et al., 2003; Xu et al., 2006, 2007; Luo et al., 2007, 2008a, 2008b, 2009a, 2009b; Safani et al., 2006; Song et al., 2007; Zeng et al., 2007, 2011)

然而淺地表介質(zhì)，通常都由富含孔隙的固體構(gòu)成，孔隙中充填流體.這種介質(zhì)的不均勻性會使波動傳播顯示出更復(fù)雜的特征.含流體孔隙介質(zhì)理論較彈性理論更精確地揭示波動在真實介質(zhì)中的傳播規(guī)律.Biot提出的多相介質(zhì)理論(Biot, 1941, 1856, 1962)將固體和流體耦合的宏觀效應(yīng)作為描述波動傳播的物理機制.理論預(yù)言的慢P波隨后被實驗證實(Plona, 1980).該理論也成為研究孔隙介質(zhì)中波動特征的基石.對于孔隙介質(zhì)中的體波，近幾十年已有較充分的論述(Carcione, 2007)，但是面波源于自由表面體波的干涉，傳播特征更復(fù)雜，研究相對缺乏，限制了實際應(yīng)用.故研究地震探測頻帶內(nèi)孔隙介質(zhì)中面波頻散、衰減，以及動態(tài)響應(yīng)特征是發(fā)展顧及含流體孔隙介質(zhì)特征高精度面波方法的關(guān)鍵.

本文從幾個方面回顧我們團隊近幾年在含流體孔隙介質(zhì)中面波傳播特征以及應(yīng)用的研究，為進一步在實際面波探測中引入多相孔隙介質(zhì)模型提供參考.

2 流體自由表面的邊界效應(yīng)

Deresiewicz和Skalak(1963)發(fā)現(xiàn)若自由表面存在流相，界面上孔隙疏通性的不同，存在“開放孔隙”，“閉合孔隙”，以及部分疏通多種邊界條件.面波傳播會受到多種流體自由表面邊界效應(yīng)的影響.基于Biot流體飽和模型的面波有零碎的研究.如Jones(1961)研究開放孔隙下孔隙介質(zhì)半空間中Rayleigh面波，發(fā)現(xiàn)流體無黏性時僅存的一種無頻散的面波.Deresiewicz(1962)進一步考慮流體黏滯性，面波發(fā)生頻散，速度在一定頻率下存在局部極小.Paul(1976a, 1976b)分析在自由表面穩(wěn)態(tài)載荷的情況下包含面波的淺地表孔隙介質(zhì)的位移響應(yīng).Tajuddin(1984)也基于無黏性流體，調(diào)查自由表面開放孔隙和閉合孔隙對面波傳播的影響.Halpern和Christiano(1986)分析自由表面諧振下孔隙介質(zhì)的動態(tài)響應(yīng).Philippacopoulos(1988)提出孔隙介質(zhì)的Lamb問題，通過4個位移勢函數(shù)獲得開放孔隙下三維動態(tài)Green函數(shù).Senjuntichai和Rajapakse(1994)研究相應(yīng)的二維問題.國內(nèi)學(xué)者王立忠等(1996)和楊峻等(1997)簡化Biot模型，計算出穩(wěn)態(tài)激振下的半空間孔隙介質(zhì)的動態(tài)響應(yīng).夏唐代等(1998, 2004)提出忽略固流耦合的慣性效應(yīng)的孔隙土力學(xué)模型，研究其中面波的頻散和應(yīng)力響應(yīng)特征.劉凱欣和劉穎(2003)考慮孔隙介質(zhì)各向異性對面波傳播的影響.國內(nèi)學(xué)者僅僅考慮開放孔隙條件.近年來也有學(xué)者發(fā)現(xiàn)閉合孔隙下，存在兩種類型面波(Edelman and Wilmanski, 2002; Edelman, 2004; Albers and Wilmanski, 2005; Gerasik and Stastna, 2014).鑒于淺地表存在疏松土層、瀝青路面，對應(yīng)開放孔隙和閉合孔隙，以及兩種極端情況之間更普遍的部分疏通條件，充分研究流體自由表面的邊界效應(yīng)對面波傳播的影響很有必要.

除與彈性介質(zhì)一樣的固相應(yīng)力邊界條件(Carcione, 2007)，流體自由表面邊界條件可寫成(Deresiewicz and Skalak, 1963)

?twz(x,y,z,t)|z=0-p(x,y,z,t)|z=0=0,

(1)

選取一組典型孔隙介質(zhì)參數(shù)(見表1)，對應(yīng)典型的含孔隙砂礫或礁石堆積物完全水飽和.其中λ和μ是骨架的拉梅常數(shù)，α和M是Biot參數(shù)，ρ和ρf分別是介質(zhì)和流體的密度，η和k分別是流體的動態(tài)黏滯度和介質(zhì)的宏觀滲透率.

表1 孔隙介質(zhì)模型參數(shù)Table 1 Model properties of a porous medium

圖1 三種流體自由邊界下表1孔隙介質(zhì)中兩類面波的相速度和衰減系數(shù)頻散，速度和衰減都由干骨架中快P波——P1波參數(shù)歸一化，參照的體波也繪制出(a) R1 波；(b) R2波.Fig.1 Phase velocities and attenuation coefficients of two surface wave modes in porous media corresponding to Table 1 under three fluid free surface boundaries. The velocity and attenuation is normalized by P1 wave in dry frame(a) R1 wave; (b) R2 wave.

2.1 固流耦合系數(shù)對流體自由表面邊界效應(yīng)的影響

固流耦合系數(shù)b控制孔隙介質(zhì)的特征頻率，隨著流體的黏滯度或骨架的滲透率的變化而變化(b=η/k).圖2所示R1波在其他參數(shù)不變，不同b時的特征.R1波頻散隨b的增大向高頻移動.b較小時，骨架和流體的黏滯摩擦較小.因而體波的衰減會在更低頻率下松弛，介質(zhì)特征頻率較低，反之亦然.這一特征也在面波中體現(xiàn).R1波的衰減系數(shù)不隨固流耦合系數(shù)的增大單調(diào)增加(圖2右).這由于衰減系數(shù)f2和f1頻率依賴的轉(zhuǎn)化疊合特征頻率增高的綜合作用.非物理模式R1波隨b增大在更高頻出現(xiàn).R2波隨b增大，f1/2耗散頻帶擴大(圖3).當(dāng)b=9.6×104Pa·s/m2時，部分疏通下R2波在大于約65 Hz出現(xiàn)負(fù)衰減，也出現(xiàn)非物理波形式輻射(圖3a右).其余條件下R2波的速度和衰減只比閉合孔隙時略小(圖3b—3d).與R1波不同的是，R2波隨b增大，速度單調(diào)降低，衰減系數(shù)單調(diào)升高.

進一步計算在垂直于自由表面應(yīng)力源下面波的地震動響應(yīng)，集中力點震源方程為

τzz(x,y,z,t)|z=0=-F(t)δ(x)δ(y),

(2)

τzz為孔隙介質(zhì)垂向正應(yīng)力.震源作用幅度為1N，時間函數(shù)F(t)為Ricker子波，主頻和時移分別是35 Hz和0.1 s.圖4所示表1介質(zhì)參數(shù)，375 m偏移距下，圖2—3中對應(yīng)5種b時面波垂直位移地震動響應(yīng).R1波振幅和相位的差異表明其受流體自由邊界效應(yīng)的影響，而快P波——P1波則差異很小.但是隨b增大，R1波的差異性減弱.振幅和相位的特征也反映相速度和衰減系數(shù)在震源子波頻帶內(nèi)的特性.R2波在多數(shù)情況下不可見，因為震源頻帶大部分落入f1/2強耗散區(qū).只有當(dāng)b=9.6×104Pa·s/m2時，R2波出現(xiàn).需注意的是，部分疏通時R2波振幅甚至大于R1波.這時R2波衰減系數(shù)更小，大于65 Hz甚至出現(xiàn)負(fù)衰減(圖3a右).可能存在R1波向R2波的輻射，這也可解釋閉合孔隙和部分疏通的R1波衰減系數(shù)相近(圖2a)，但部分疏通時的R1波振幅更弱(Zhang et al., 2011).

2.2 孔隙迂曲度對流體自由表面邊界效應(yīng)的影響

Berryman(1980)給出由孔隙度估計孔隙迂曲度的公式，

C=1-r0(1-1/φ),

(3)

其中r0為固體顆粒形狀參數(shù)，r0=1/2代表球狀顆粒，其他橢球體r0取值0～1.近似地，一般我們?nèi)∏驙铑w粒時的迂曲度.圓柱狀孔隙且平行于流體壓力梯度，C=1，為迂曲度最小值，隨機孔隙分布，C=3(Mavko et al., 1998).圖5所示R1波在迂曲度為1、2、3時頻散的相速度和衰減系數(shù).C較小時R1波頻散更明顯.這由于流體流動受迂曲度的影響，迂曲度較小時，流體流動受阻礙更小，固流相互作用更大，衰減得以顯現(xiàn).C變化時，開放孔隙時R1波的速度和衰減變化幅度最小，對流體流動變化最不敏感.圖6所示R2波在迂曲度為1、2、3時頻散的相速度和衰減系數(shù).兩種情況下R2波有相似特征，C較大時頻散的幅度會減小，類似R1波.隨C變大，部分疏通下R2波與閉合孔隙下R2波的衰減系數(shù)的分離出現(xiàn)在更低頻率，差異也更明顯.

圖7所示與圖4相同震源和接收參數(shù)時，迂曲度為1、2、3時面波地震動響應(yīng)特征.孔隙迂曲度較小時，R1波振幅最弱，隨著C的增加，R1波振幅增強.參照P1波，迂曲度變化時，三種流體邊界下波形和振幅差異較小，說明P1波受孔隙流體的影響較R1波小.R2波沒有觀察到.

3 移動震源激發(fā)效應(yīng)

由于淺地表巖石的固結(jié)程度低，某些軟弱土層中面波速度很低.這時，震源激發(fā)的速度會對面波地震動響應(yīng)造成明顯的影響.含流體孔隙介質(zhì)在移動震源激發(fā)下的動態(tài)響應(yīng)是重要的工程問題，如Burke和Kingsbury(1984)給出在地表移動壓力下孔隙介質(zhì)響應(yīng)的解析解.Siddharthan等(1993)忽略固流耦合，分析平面應(yīng)變下移動載荷下多層孔隙介質(zhì)響應(yīng).Jin等(2004)獲得地表勻速線載荷下的二維介質(zhì)響應(yīng)的半解析表達.Theodorakopoulos(2003), Theodorakopoulos等(2004)通過移動線載荷的解析和近似數(shù)值方法研究二維土壤介質(zhì)的響應(yīng).Lu和Jeng(2007)首次推導(dǎo)勻速點載荷下的三維孔隙介質(zhì)的動態(tài)響應(yīng).Xu等(2008)進一步擴展到多層的情況.Cai等(2007, 2008)研究有限矩形載荷和軌道系統(tǒng)下，骨架不可壓縮時三維穩(wěn)態(tài)響應(yīng).Cao和Bostr?m (2013)研究加速和減速的火車載荷軌道系統(tǒng)激發(fā)的孔隙介質(zhì)響應(yīng)特征.Lefeuve-Mesgouez和Mesgouez (2008, 2012)通過波模態(tài)分析半空間及多層孔隙黏彈性介質(zhì)在高速諧振矩形載荷下的地震動傳播過程.Li等(2012)分析孔隙介質(zhì)的波數(shù)域近似解表示移動諧振下半空間響應(yīng)的精度.Beskou和Theodorakopoulos(2011)全面回顧移動激發(fā)問題.穩(wěn)態(tài)載荷(單色頻率)過于簡單，不能有效利用振動響應(yīng)包含面波傳播的有效信號.將移動載荷視為有限頻率噪聲源，并考慮流體自由表面邊界效應(yīng)，分析激發(fā)的面波傳播特征可以更充分認(rèn)識移動震源效應(yīng)的影響.

圖2 不同固流耦合系數(shù)時R1波的相速度和衰減系數(shù)頻散Fig.2 Phase velocity and attenuation coefficient of R1 wave for different solid-fluid coupling coefficients

圖3 不同固流耦合系數(shù)時R2波的相速度和衰減系數(shù)頻散Fig.3 Phase velocity and attenuation coefficient of R2 wave for different solid-fluid coupling coefficients

圖4 不同固流耦合系數(shù)時面波的地震動響應(yīng)Fig.4 Seismic responses of surface wave for different solid-fluid coupling coefficients

圖5 不同孔隙迂曲度時R1波的相速度和衰減系數(shù)頻散Fig.5 Phase velocity and attenuation coefficient of R1 wave for different tortuosities

圖6 不同孔隙迂曲度時R2波的相速度和衰減系數(shù)頻散Fig.6 Phase velocity and attenuation coefficient of R2 wave for different tortuosities

圖7 不同孔隙迂曲度下面波的地震動響應(yīng)Fig.7 Seismic responses of surface wave for different tortuosities

圖8 xy平面內(nèi)以勻速v沿x運動的移動震源和接收點的相對位置Fig.8 The decomposition of the velocity of the moving impulse along x axis related to the receiver on the xy plane

軟弱孔隙介質(zhì)參數(shù)見表2，對應(yīng)某一高速公路施工現(xiàn)場采集的未固結(jié)砂巖樣本.該樣本有較低的剪切模量和較高的滲透率，反映淺表含水層的典型地質(zhì)條件.流體對應(yīng)淺地表水的狀況.選擇沿x軸方向以速度v移動的地表點應(yīng)力震源，可以寫成

τzz(x,y,z,t)|z=0=-F(t)δ(x-vt)δ(y).

(4)

對比方程(3)，加入震源速度的擾動，作坐標(biāo)參數(shù)替換x′=x-vt.震源幅度為100N，震源過坐標(biāo)原點時開始計時，時間響應(yīng)仍為Ricker子波，時移0.05s，接收點在(30m, 30m, 1m)處(圖8).選擇低主頻5Hz震源突出震源移動的影響(Zhangetal., 2014a).圖9所示三種流體自由表面(開放孔隙、閉合孔隙以及滲透剛度0.1時部分疏通情況)下垂直位移和孔隙流體壓力的面波地震動響應(yīng).表2對應(yīng)等效Gassmann介質(zhì)Rayleigh波速度約為169m·s-1，移動震源速度選擇從亞Rayleigh波速度(40m·s-1)到超Rayleigh波速度(200m·s-1).只有P1和R1波可以分辨.震源速度增大到169m·s-1，所有波形明顯壓縮.震源速度進一步增大到200m·s-1，這些現(xiàn)象更為明顯.這源于Doppler藍移，由徑向視速度產(chǎn)生(圖8).圖10比較開放孔隙時垂直位移和流體壓力在震源速度v為40 m·s-1和200 m·s-1時功率譜.相比v=40 m·s-1，v=200 m·s-1時頻率成分有明顯的上移，無論垂直位移還是流體壓力，主頻甚至大于10 Hz，比震源原始主頻5 Hz大1倍.移動震源的Doppler效應(yīng)對面波傳播頻率響應(yīng)有明顯影響.波形扭曲出現(xiàn)，特別是R1波，表現(xiàn)為子波拖長.這源于橫向速度(圖8).主頻較低(5 Hz)，子波時長大概0.2 s(T≈1/5=0.2 s)，移動震源激發(fā)時有明顯的橫向移動.

4 多相流體效應(yīng)

孔隙中流體的特性會極大地影響介質(zhì)中波動傳

表2 軟弱孔隙介質(zhì)物性參數(shù)Table 2 Model properties of a weakly porous medium

播.通常情況，孔隙空間內(nèi)通常是氣體和液體混合充填的，這時在地震頻帶內(nèi)Biot宏觀固流耦合效應(yīng)難以解釋觀測到的衰減.有研究表明(Pride et al., 2004)，中觀尺度的流體斑塊飽和是造成波動衰減的主要因素.這種機制由于中觀尺度下波動激發(fā)的流體流動造成.White(1975)最早建立分離的球狀氣體飽和斑塊和液體飽和背景模型來描述這一特征.波動造成的流體壓力梯度使氣飽和斑塊擴散形變，驅(qū)動斑塊和背景間的流體對流，導(dǎo)致介質(zhì)的本征衰減(Johnson, 2001; Vogelaar et al., 2010).這一低頻的部分飽和效應(yīng)已在體波研究中得到廣泛重視.觀測資料和數(shù)值模擬也與這種斑塊飽和模型一致(Wenzlau and Müller, 2009)，波動激發(fā)的對流改變介質(zhì)整體的頻率響應(yīng)(Saenger et al., 2009).面波由體波干涉而成，斑塊飽和這一機制必然會影響面波.進一步理解斑塊飽和對面波的影響可以使我們更精確估計面波波速和衰減，建立更精確反演成像模型.

中觀尺度(典型值為數(shù)十厘米)遠(yuǎn)大于孔隙尺度，又遠(yuǎn)小于波長.因此，在中觀尺度上，波動激發(fā)的流體壓力變化在一個波動周期內(nèi)不能完全平衡.非均勻的斑塊飽和經(jīng)常產(chǎn)生于氣液接觸區(qū)域.在完全氣飽和及完全液飽和區(qū)域之間，通常會有一轉(zhuǎn)換區(qū)域存在.波動激發(fā)的對流會在這一區(qū)域內(nèi)造成壓力梯度，并驅(qū)使氣流接觸面的擴散效應(yīng)(Pride and Berryman, 2003).

(5)

整個介質(zhì)在低頻極限下或稱之靜態(tài)極限下的等效體積模量，可以寫成

KGW=Km+α2M(Kw),

(6)

其中Km是骨架的體積模量，KGW是等效流體孔隙介質(zhì)的Gassmann體積模量.

頻率足夠高，波周期內(nèi)流體不能完全松弛，造成流體的壓力梯度.區(qū)域內(nèi)的流體壓力不一致，但仍可假設(shè)各個流相中的壓力一定.介質(zhì)的體積模量可以通過Hill理論給出.

(7)

或者Voigt等效流體的Gassmann體積模量給出

KGV=Km+α2M(KV),

(8)

其中

KV=∑SiKfi,

(9)

忽略對流效應(yīng)，也稱之為非流動極限(Mavkoetal., 1998).

在兩極限頻率之間的轉(zhuǎn)換反映斑塊飽和模型中波動的頻率響應(yīng)，體現(xiàn)中觀機制的影響.White(1975)的處理方法(Dutta和Seriff(1979)修正)獲取這一模型復(fù)頻率效應(yīng).Johnson(2001)推廣White模型得到因果性的近似廣義解.Vogelaar等(2010)進一步給出精確解.

斑塊飽和介質(zhì)性質(zhì)如表3所示.同心球殼的外半徑為b=20 cm，代表斑塊的尺度.整個斑塊飽和介質(zhì)源自XH高速公路附近采集的空氣-水飽和未固結(jié)含砂粘土樣本.

圖12a顯示多相流體斑塊飽和介質(zhì)中頻散的P1波相速度和衰減(1/Q).其中氣體占流體比例為0.3.在低頻端P1波接近靜態(tài)極限；在高頻端P1波接近非流動極限.兩極限之間的轉(zhuǎn)換在地震頻帶內(nèi)顯現(xiàn)(1～200 Hz)，這也解釋了這一頻帶內(nèi)波動傳播的主要衰減.可以發(fā)現(xiàn)方程fm=Kf2k/2πη2φ(b-a)2(圖中垂直點線)給出近似的轉(zhuǎn)換特征頻率 (DuttaandSerriff, 1979)，這一頻率也區(qū)分松弛和非松弛狀態(tài).但需要注意，這一機制在Biot理論框架內(nèi)對S波影響較小，因為兩種極限對于S波來說相互重合.這一結(jié)論，亦被數(shù)值模擬驗證(Rubinoetal., 2009;Liuetal., 2009a).

圖9 不同震源移動速度下固體垂直位移和孔隙流體壓力的面波地震動響應(yīng)Fig.9 Seismic responses of solid vertical displacement and pore fluid pressure for different source velocities

圖10 震源移動速度為40 m·s-1和200 m·s-1時固體垂直位移(a)和孔隙流體壓力(b)面波地震動的功率譜Fig.10 Power spectra of solid vertical displacement (a) and pore fluid pressure (b) of surface wave seismic responses for source velocities at 40 m·s-1 and 200 m·s-1

圖11 由周期性的半徑為a，相隔距離為2b′的球狀氣體飽和區(qū)域構(gòu)成的立方體格子，改自Vogelaar et al., 2010每一氣體飽和區(qū)域由一個外半徑為b的液體飽和的球殼包圍， 所以立方體格子的體積等于同心球體的體積，即Vb′= Vb.Fig.11 Geometry of the cubic lattice of periodic spherical gas pocket with radius a, separated by distance 2b′ (from Vogelaar et al., 2010)Each gas pocket is surrounded by a liquid shell with radius b, so that the volume of the cube equals to the volume of the sphere Vb′=Vb.

宏觀上描述該模型的自由表面邊界效應(yīng)，可以用等效復(fù)體積模量表征的邊界條件來獲取Rayleigh面波的久基方程和地震動動態(tài)響應(yīng)(Zhangetal., 2014b).選擇和體波分析一致的介質(zhì)模型，面波頻散的相速度和衰減如圖12b所示.斑塊飽和衰減同樣在低頻段的面波頻散中顯現(xiàn)，與P1波類似(圖12a).轉(zhuǎn)換也在兩個頻率極限——靜態(tài)極限和非流動極限之間出現(xiàn)(圖12b左).衰減1/Q較水飽和和氣飽和下明顯增強.對應(yīng)最大1/Q轉(zhuǎn)換特征頻率在數(shù)十赫茲內(nèi)顯現(xiàn)，落入地震頻帶內(nèi)，也與P1波類似(圖12a右).因為S波受流體分布影響較小，面波的這一趨勢主要反映P1的影響.雖然面波相對P1波的敏感性不如S波顯著(Xia et al., 1999)，但低頻段內(nèi)P1波的影響不能被忽略.Biot機制特征頻率很高，可以推斷在低頻帶內(nèi)，斑塊飽和衰減機制對面波的衰減有顯著的影響.因而必須小心建立面波的衰減模型，同時強調(diào)頻率依賴和P1波的影響.

面波的地震動響應(yīng)也反映這一特征(圖13)，其中震源為Ricker子波，主頻35 Hz，時移0.1 s，幅度100 N，偏移距200 m.可以看到顧及斑塊飽和機制的面波如P1波一樣，比水飽和氣飽和僅有Biot宏觀機制時振幅明顯衰減.

數(shù)值實驗還發(fā)現(xiàn)，這一中觀尺度上的面波衰減與氣體在流體中的比例有很明顯的關(guān)系，也為利用面波探測地下流體分布提供可能(Zhang et al., 2014b).

表3 斑塊飽和孔隙介質(zhì)的物性參數(shù)Table 3 Model properties of a patchy saturated porous medium

5 面波傳播的等效特征

P2波在低頻段速度低，且具有很強的擴散性質(zhì)，造成Biot波動方程數(shù)值計算時很不穩(wěn)定，需要很大的存儲量和計算時間(Carcione and Quiroga-Goode, 1995, 1996).為克服這些困難，找到孔隙介質(zhì)中波動傳播的等效解是有效手段.應(yīng)用黏彈性等效單元模擬宏觀固流耦合效應(yīng)研究體波傳播最早見于Geertsma和Smit(1961)的研究.P1波的品質(zhì)因子(Q)滿足Q>5，孔隙介質(zhì)中的衰減可用一單元的標(biāo)準(zhǔn)線性體(Zener體)等效表示(Ben-Menahem and Singh, 1981).Carcione(1998)成功地應(yīng)用Zener等效固體表征Biot宏觀流和噴流機制(Dvorkin et al., 1994).并成功模擬包含P波和S波的等效波場，建立記憶變量等效代替.雖然Biot的宏觀流機制不涉及固體的體形變，但是用Zener模型，可直接匹配每一種波型的頻散和衰減(Carcione, 1998).這種等效模型也用來描述波動散射的問題(Morochnik and Bardet, 1996).最近的研究表明，這種等效模型也可以用來表述部分飽和(Rubino et al., 2009; Picotti et al., 2010, 2012a; Carcione et al., 2012)、雙孔隙(Liu et al., 2009b, 2010)，以及多相固體等提升Biot模型(Picotti et al., 2012b).這些研究表明，復(fù)雜介質(zhì)可用簡單的等效介質(zhì)表征，等效黏彈性固體中用松弛函數(shù)直接匹配每一種波型可以有效地表征孔隙介質(zhì)中波動傳播的宏觀特征(Carcione, 2007).由于自由表面的存在，Rayleigh面波的數(shù)值計算方法面臨更為嚴(yán)苛的不穩(wěn)定問題，即便是在完全彈性固體中，需要在很小的網(wǎng)格中求解差分方程(Xu et al., 2007; Zeng et al., 2011).流體自由表面會使面波特征更為復(fù)雜(Zhang et al., 2011, 2012)，數(shù)值計算的穩(wěn)定性更差.這也為實際面波應(yīng)用中引入多相孔隙模型帶來困難.因此，在前人用等效固體表征體波的同時，我們也需要回答包含自由界面效應(yīng)的面波是否也可用等效黏彈性固體表征.

圖12 P1波(a)和R1波(b)在斑塊飽和模型中的相速度和衰減頻散特征.S1=0.3，b=0.2 m上下水平線表示兩極限頻率下的體積模量KGW和KGH.垂直的點線近似表示轉(zhuǎn)換特征頻率.R1波衰減的對照組代表氣飽和和水飽和時的情況.Fig.12 Phase velocities and attenuation of P1 wave (a) and R1 wave (b) in the patchy saturation model. S1=0.3, b=0.2 m The lower and upper horizontal lines are obtained from limiting modulus KGW and KGH. The vertical dot line is the approximating transition frequency. The control groups for R1 attenuation represent the half space fully saturated by air and water.

圖13 面波在斑塊飽和模型中的地震動響應(yīng).S 1=0.3，b=0.2 m 對照組代表氣飽和水飽和時的情況.Fig.13 Seismic responses of surface wave in the patchy saturation model. S1 = 0.3 and b=0.2 mThe control groups represent the half space fully saturated by air and water.

基于含流體孔隙介質(zhì)模型，等效黏彈性模型可由黏彈性單元替代原本構(gòu)方程中的復(fù)模量實現(xiàn)(Carcione, 2007).在單相的黏彈性模型中，體波可以等效地用速度和衰減相關(guān)的松弛模式等效表征.Rayleigh面波也可以用包含兩種體波松弛模式的黏彈性固體結(jié)合邊界條件實現(xiàn).

使用標(biāo)準(zhǔn)線性體模型(Zener模型)來表征快波(P1波和S波)，Zener黏彈性單元的復(fù)模量可以表示成(Carcione, 2007)

(10)

其中M0是介質(zhì)松弛模量，τε和τσ分別是應(yīng)力和應(yīng)變松弛時間.每一種體波，只需一種黏彈性單元就足以表征體波在孔隙介質(zhì)中的行為.因此，每一體波只需要一組松弛時間τε和τσ來表征.在Zener模型中，松弛時間可表示成

(11)

其中ω0=2πf0對應(yīng)波動衰減峰值頻率，Q0是該頻率f0下的品質(zhì)因子.在等效介質(zhì)中，孔隙流體的自由表面效應(yīng)被忽略，整個介質(zhì)的衰減由黏彈性固體的松弛機制代替.等效的表征可以簡單替換(11)中的對應(yīng)快波的模量，松弛時間以相應(yīng)P1波和S波的f0i和Q0i,i=P,S求得，同時忽略P2波的影響.等效波動方程可通過Biot方程中對應(yīng)等效替換并去掉P2波項得到(Zhang et al., 2014c).

面波的等效相速度和1/Q亦可通過求解等效介質(zhì)中Rayleigh面波的久基方程求得，相應(yīng)的面波地震動響應(yīng)可以通過類似的積分變換方法得到.

圖14 面波在未固結(jié)砂巖中相速度(左)和衰減(右)的頻散孔隙中充填水(a)和空氣(b).兩種等效黏彈性固體中快波的衰減峰值頻率和品質(zhì)因子分別為436.9 Hz, 421.0 Hz 和 17.5, 13.3 (a)，以及12488.5 Hz, 5840.9 Hz和4488.5, 9200.6 (b).孔隙含水的情況下，等效介質(zhì)在低頻段低估面波的衰減.Fig.14 The dispersion of phase velocity (left) and attenuation (right) of surface wave in the unconsolidated sand for water (a) and air (b) saturated models The two equivalent-viscoelastic materials have relaxation frequency of 436.9 Hz, 421.0 Hz and quality factor of 17.5, 13.3 for the two body wave modulus for the water case (a), and relaxation frequency of 12488.5 Hz, 5840.9 Hz and quality factor of 4488.5, 9200.6 for the air case (b), respectively. The effective representation underestimated the attenuation for the water case at low frequencies.

圖15 面波在未固結(jié)砂巖中的等效地震動響應(yīng)波形孔隙中充填水(a)和空氣(b).震源激發(fā)的主頻分別為5 Hz(左)、50 Hz(中)和200 Hz(右).除50 Hz和200 Hz孔隙含水時(a)，等效介質(zhì)中波形顯示出與孔隙介質(zhì)相當(dāng)好的一致性.Fig.15 Seismic responses of surface wave in the unconsolidated sand for water (a) and air (b) saturated models The source pulses have center frequency of 5 Hz (left), 50 Hz (center), and 200 Hz (right). Except for 50 Hz and 200 Hz for water case (a), the wave forms show a fairly well match with the fluid-saturated model.

圖15所示未固結(jié)砂巖中面波的垂直位移地震動響應(yīng).接收點位于自由表面，偏移距為100 m.時移0.1 s的Ricker子波震源的主頻fs分別為5 Hz(左)、50 Hz(中)和200 Hz(右).在孔隙含水時(圖15a)，震源主頻為50 Hz和200 Hz時，等效黏彈性介質(zhì)未能擬合孔隙介質(zhì)中的波形.而在fs=5 Hz則顯示出相當(dāng)好的擬合效果.而在孔隙含空氣時(圖15b)，所有結(jié)果都顯示出較好的一致性.

含多相流體的孔隙介質(zhì)，由于P1、P2波的共同作用造成中觀尺度下對流，引起波傳播的衰減，也可以直接對等效體積模量以及S波做黏彈性等效替換，得到類似的等效表征模型(Zhang et al., 2014d).

6 結(jié)論及展望

精確認(rèn)識面波在真實介質(zhì)傳播時的速度和衰減特征是提高面波方法勘探精度的基礎(chǔ).本文從面波的幾個方面詳細(xì)討論了面波在更接近實際介質(zhì)的孔隙介質(zhì)模型中的傳播特性，獲得以下幾點認(rèn)識：

(1)流體在自由表面不同行為造成面波傳播的差異使得在面波的數(shù)據(jù)采集和處理時，根據(jù)不同地表條件引入流體自由表面效應(yīng)的修正.特別是針對淺地表巖石固結(jié)程度低、滲透率較高的情況.有些特殊情況甚至要考慮第二模式面波的影響.這種修正的方法和評價標(biāo)準(zhǔn)，需要再結(jié)合實際情況進一步的研究.

(2)針對淺地表介質(zhì)軟弱，面波速度較低的情況，震源激發(fā)時移動速度有時候也需要評價.相對于觀測點的徑向、橫向移動會造成面波信號頻率響應(yīng)的明顯變化，特別是震源激發(fā)頻率較低時.在有著較高速度移動噪聲源的環(huán)境中，這種效應(yīng)的影響需要根據(jù)采集觀測系統(tǒng)和主要噪聲源的方位判斷.擾動的影響的大小需要根據(jù)實際采集環(huán)境謹(jǐn)慎地評價.

(3)當(dāng)介質(zhì)部分飽和時，中觀尺度多相流體的斑塊飽和效應(yīng)會在地震探測頻帶內(nèi)顯著影響面波的傳播.這一機制主要影響脹縮運動的P波，故而在地震頻帶內(nèi)估計面波的衰減，需要重新重視P波的影響.

(4)根據(jù)面波的傳播特征建立等效黏彈性模型，在地震探測頻帶內(nèi)能較好地表征面波在孔隙介質(zhì)中的傳播，這有利于在面波的數(shù)值計算中更經(jīng)濟地引入孔隙介質(zhì)模型，建立更少參數(shù)約束的面波反演成像模型.

可以發(fā)現(xiàn)，含流體的孔隙介質(zhì)中面波的傳播較彈性介質(zhì)更為復(fù)雜.在面波實際應(yīng)用中顧及復(fù)雜傳播特性影響的研究尚在起步階段，如何更合理有效地引入孔隙介質(zhì)模型，有更多的理論實際問題需要進一步探討.

致謝 感謝匿名審稿人對本文的建議.

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(本文編輯 何燕)

Characteristics and application of surface wave propagation in fluid-filled porous media

ZHANG Yu1,2,3,4, XU Yi-Xian4,5, XIA Jiang-Hai4, ZHANG Shuang-Xi1,2,3, PING Ping6,4

1SchoolofGeodesyandGeomatics,WuhanUniversity,Wuhan430079,China2KeyLaboratoryofGeospaceEnvironmentandGeodesy,MinistryofEducation,Wuhan430079,China3CollaborativeInnovationCenterofGeospaceInformationScience,WuhanUniversity,Wuhan430079,China4SubsurfaceMulti-scaleImagingLaboratory,ChinaUniversityofGeosciences,Wuhan430074,China5StateKeyLaboratoryofGeologicalProcessesandMineralResources,ChinaUniversityofGeosciences,Wuhan430074,China6StateKeyLaboratoryofGeodesyandEarth′sDynamics,InstituteofGeodesyandGeophysics,ChineseAcademyofSciences,Wuhan430074,China

High frequency surface wave method based on seismic wave propagation in single-phase media has been widely applied for acquiring near surface shear wave velocity in several fields. However, the near surface earth media, consolidated and unconsolidated, for the general hydrogeological conditions, bear plenty of pores. Fluid in pores affects the surface wave propagation in the media remarkably, which is represented in dispersion in velocity and attenuation.

國家重點基礎(chǔ)研究計劃(2013CB733303)，國家自然科學(xué)基金(41304077, 40974079)，中國博士后科學(xué)基金(2014T70740, 2013M531744)，教育部地球空間環(huán)境與大地測量重點實驗室(12-02-03)和湖北省多尺度地下成像重點實驗室(SMIL-2014-01)聯(lián)合資助.

張煜，1982年生，博士，主要從事淺地表地球物理學(xué)相關(guān)的研究.E-mail：yuz124@gmail.com

10.6038/cjg20150812

P631

2014-12-22，2015-06-11收修定稿

10.1007/s00161-005-0203-y.

張煜, 徐義賢, 夏江海等. 2015. 含流體孔隙介質(zhì)中面波的傳播特性及應(yīng)用.地球物理學(xué)報，58(8):2759-2778,doi:10.6038/cjg20150812.

Zhang Y, Xu Y X, Xia J H, et al. 2015. Characteristics and application of surface wave propagation in fluid-filled porous media.ChineseJ.Geophys. (in Chinese),58(8):2759-2778,doi:10.6038/cjg20150812.