Pore pressure coefficient for soil and rock and its relation to compressional wave velocity

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1 Title Poe pessue coefficient fo soil and ock and its elation to compessional wave velocity Autho(s) Yang, J Citation Geotechnique, 25, v. 55 n. 3, p Issued Date 25 URL Rights Geotechnique. Copyight Thomas Telfod Ltd.; This wok is licensed unde a Ceative Commons Attibution- NonCommecial-NoDeivatives 4. Intenational License.; Pemission is ganted by ICE Publishing to pint one copy fo pesonal use. Any othe use of these PDF files is subject to epint fees

2 Yang, J. (25). Géotechnique 55, No. 3, TECHNICAL NOTE Poe pessue coefficient fo soil and ock and its elation to compessional wave velocity J. YANG* KEYWORDS: compessibility; dynamics; laboatoy tests; patial satuation; poe pessues INTRODUCTION In laboatoy tiaxial tests it has been common to check the specimen s degee of satuation by detemining the poe pessue coefficient, B. The method is known as the B-value test and basically oiginated fom Skempton s wok on evaluation of the poe pessues in clays in the design of eath dams, in which he deived the well-known expession fo B (Skempton, 954): B ¼ u ó ¼ () þ nk ð b =K f Þ Hee ó is a small incease in all-ound o confining pessue applied to an element of soil in an eath dam o to the sample in the tests, and u is the esulting change in poe pessue measued unde undained conditions. The poosity of soil is denoted in equation () by n, denotes the bulk modulus of the soil skeleton, and K f the bulk modulus of the mixtue of poe wate and ai. Since the compessibility of wate is negligible compaed with that of the soil skeleton, equation () indicates that B fo satuated soils, wheeas fo patially satuated soils, B,, with a typical ange of. to. 5 at the optimum wate content. In deiving equation () the solid gains wee assumed as incompessible, which has been a widely accepted assumption in soil mechanics, but may not be applicable in ock mechanics. Fo soils and ocks with compessible paticles, Bishop (973) showed late on that the poe pessue coefficient was expessed by B ¼ u ó ¼ (2) þ nð=k w =K s Þ=(= =K s ) in which K s is the bulk modulus of solid paticles. Recently, Kokusho (2) pesented the following expession fo B using Biot s fomulation given by Zienkiewicz & Bettess (982): B ¼ = =K s = þ n=k f n=k s (3) It is not difficult to note that equation (3) diffes fom Bishop s expession when the latte is witten in an altenative fom as Manuscipt eceived 24 Febuay 24; evised manuscipt accepted 2 Septembe 24. Discussion on this pape closes on 3 Octobe 25, fo futhe details see p. ii. * Depatment of Civil Engineeing, The Univesity of Hong Kong. B ¼ = =K s = þ n=k f ( þ n)=k s (4) Since Biot s theoy descibes the behaviou of poous media such as soils and ocks moe igoously, a concen natually aises ove the coectness of eithe Bishop s o Kokusho s expession, as well as the cause leading to the diffeence between them. It is the pupose of the pesent study to claify this issue, both analytically and numeically. Besides the poe pessue paamete B, a new indicato of satuation fo soils the velocity of compessional waves (i.e. P-waves), V p has been advocated, and has eceived inceasing attention (Yang & Sato, 998, 2a; Kokusho, 2; Ishihaa et al., 2). Compaed with the poe pessue paamete B, the P-wave velocity can be measued conveniently in the field and thus has an advantage in pactical applications. The effectiveness of the use of P-wave velocity in identifying in-situ patially satuated zones has been well demonstated by a boehole aay site (Yang & Sato, 2a). Moe ecently, an application of P-wave velocity in intepeting laboatoy test data fo the cyclic stength of patially satuated sands has been poposed by Yang (22), who showed a useful elation between B and V p : V p ¼ 4ì=3 þ K =2 b=( B) (5) in which ì (o G) is the shea modulus of the soil skeleton and is total mass density. In deiving equation (5) the compessibility of solid gains was not of pimay concen and theefoe not included in the expessions fo eithe B o V p. In this study a geneal elation between B and V p will be established within the famewok of Biot s theoy, and the effect of paticle compessibility will be identified numeically. This is a second pupose of the pesent study. DERIVATION OF PORE PRESSURE PARAMETER The analysis is based on classical Biot s theoy (Biot, 956, 962), which models the complex inteactions between the solid and fluid pats using the macoscopic laws of mechanics. In its simplest fom the analysis equies the following assumptions: (c) (d) (e) The poous medium is statistically isotopic in such a way that fo any coss-section the same atio of the fluid aea to the solid aea applies. The void space of the poous medium is inteconnecting, and the sealed poe space is pat of the solid. Both the solid skeleton and the solid mateial foming it ae elastic and isotopic. The solid gains, solid skeleton and poe fluid ae compessible. The poe fluid is viscous, and Dacy s law govens its flow. 25

3 252 YANG It is customay to denote the displacement of the solid pat by u i and the displacement of fluid elative to the solid by w i. Thus the field equations accounting fo both inetial and viscous inteactions between the two pats in the poous medium can be given as ìu i, jj þ º þ Æ 2 M þ ì e,i ÆMæ,i ¼ u i þ f w i (6) ÆMe,i Mæ,i ¼ f u i þ f n w i þ ç k9 _w i (7) whee e ¼ u j, j and æ ¼ w j, j ; s is the mass density of solid gains, f is the mass density of fluid, and ¼ ( n) s + n f ; º is the Lamé constant of the solid fame; and Æ and M ae paametes accounting fo the compessibilities of solid and fluid constituents, and given by Æ ¼ K s (8) M ¼ K 2 s K s þ n K (9) s K f Obviously, fo incompessible solid gains Æ ¼ and /M ¼ n/k f. Note that k9 in the field equations diffes fom the pemeability coefficient k (m/s) that is used in soil mechanics. They ae elated by k9 ¼ k ç () f g in which ç is fluid viscosity and g is the gavitation acceleation at which the pemeability is measued. The field equations (6) and (7) have been obtained by enteing into the equilibium conditions the following elationships between stess, poe pessue, and stain: ó ij ¼ ºeä ij þ 2ìå ij Æä ij p f () p f ¼ Mæ ÆMe (2) whee ó ij is the stess tenso, å ij ¼ (u i, j + u j, i )/2 is the stain tenso, p f is poe pessue, and ä ij is the Konecke delta. Hee a concept of homogenisation has been intoduced that assumes that the mixtue of poe wate and ai can appoximately be teated as an equivalent homogeneous poe fluid completely filling the voids with a single poe pessue. Note that equation () implies the effective stess law in the following fom: ó ij ¼ ó 9 ij ä ij Æ p f (3) in which ó 9 ij is the effective stess tenso. Clealy, equation (3) can be educed to the well-known effective stess equation intoduced by Tezaghi if the compessibility of solid gains is ignoed. Equation (2) can eadily be ewitten as w i,i ¼ Æu i,i þ p f (4) M By specifying the undained conditions that thee is no flow of poe fluid, one aives at Æu i,i þ p f M ¼ (5) o K b ó 9ii þ Æ þ n n p f ¼ (6) K s 3 K s K f K s Intoducing the effective stess law expessed in equation (3) into the above equation leads to ó ii þ n n þ Æ p f ¼ (7) K s 3 K f K s fom which the poe pessue coefficient can eadily be deived as B ¼ 3 p f = =K s ¼ (8) ó ii ð= =K s ) þ n(=k f =K s ) Note that the negative sign is equied by the sign convention embedded in equation (). Now, it is of inteest to obseve that the expession deived above is the same as that by Bishop (973), implying that the expession due to Kokusho (2) is in eo. It is found that a majo cause of the eo was incoect use of the effective stess law: the effective stess equation by Tezaghi athe than the geneal fom in equation (3) was used, and a consistency was lost. Futhemoe, by intoducing the commonly used expession fo the compessibility of poe fluid (Yang & Sato, 998, 2a), ¼ þ S (9) K f K w p a whee S is the degee of satuation, K w is the bulk modulus of poe wate, and p a is the absolute fluid pessue, equation (8) can be witten as a function of satuation: B ¼ = =K s = ( þ n)=k s þ n=k w þ n( S )= p a (2) A GENERAL EXPRESSION FOR COMPRESSIONAL WAVE VELOCITY In the same context of Biot s theoy, the field equations (6) and (7) can be solved with the aid of potential functions, yielding two compessional waves and one shea wave. In the low-fequency ange to which the fequencies involved in most soil dynamics poblems belong, only the fast compessional wave exists, with its velocity being independent of fequency, while the slow one is a diffusion pocess owing to the viscous coupling between the solid and fluid phases. The inteested eade may efe to Yang & Sato (2b), whee moe details about the two compessional waves ae pesented, with a special efeence to the vibation of a soil column. The velocity of the fast compessional wave at low fequencies is given by (Yang & Sato, 2a)! =2 (2) V p ¼ º þ 2ì þ Æ2 M It can be veified that fo a limiting case of incompessible gains the above expession takes the fom as given in Yang (22): V p ¼ º þ 2ì þ K =2 f =n (22) Similaly, the geneal expession in equation (2) can be witten as a function of the degee of satuation as Kb þ 4ì=3 V p ¼ Æ 2 =2 þ (23) ðæ nþk s þ n=k w þ n( S )= p a Fo a fully satuated poous medium with incompessible

4 PORE PRESSURE COEFFICIENT FOR SOIL AND ROCK 253 paticles, equation (23) educes to the following simple fom: V p ¼ K =2 b þ 4ì=3 þ K w =n (24) With equations (8) and (2) in hand a geneal elationship between B and V p can be obtained: " # =2 V p ¼ 4ì=3 þ = ð ÆBÞ (25) To the autho s knowledge, equation (25) is the fist poposal of the explicit elationship between B and V p in which the paticle compessibility is consistently accounted fo in the famewok of Biot s theoy. It can be shown that in a limiting case of incompessible gains this geneal elation becomes the simple one given by Yang (22). An altenative fom fo the above expession can be given by intoducing the shea wave velocity, V s, an engineeing popety diectly linked with the shea modulus of soil and ock: B ¼ 2 3 Æ 3 4 3V 2 p 4V 5 (26) 2 s EFFECT OF PARTICLE COMPRESSIBILITY The influence of paticle compessibility, skeleton compessibility and poosity on the value of B at full satuation is illustated in Fig., whee compaison of the values given by Skempton s expession (equation ()) and by equation (8) is included. Hee the bulk modulus of solid paticles is assumed as a typical value (i.e. K s ¼ 36 GPa), and the bulk modulus of the skeleton,, is nomalised by K s. Fo puposes of compaison, the values calculated by equation (3) (Kokusho s expession) ae pesented togethe with those given by equation (8) in Fig.. It is inteesting to note fom Fig. that fo poous media of high skeleton compessibility, say,., the diffeence between the values of B given by equations () and (8) is not significant. In othe wods, whethe o not the paticle compessibility is taken account of will not impact on the pediction of B. Howeve, as the skeleton compessibility appoaches the compessibility of solid gains, majo eos will aise fom the neglect of the paticle compessibility. Indeed, a wide ange of the skeleton compessibility has been epoted fo poous o fissued ocks, fo which it is not uncommon that may be of the ode of. 5. Figue also indicates that, at a specific lage value of, the diffeence caused by neglect of the paticle compessibility is sensitive to the value of poosity. The diffeence will become much geate as the poosity deceases, with Skempton s expession giving a highe value of B. Based on the geneal expession given in equation (2), the elation between the poe pessue atio B and the degee of satuation S is plotted in Fig. 2 fo vaious values of, whee Fig. 2 is fo a poosity of.35 and Fig. 2 fo a poosity of. 5. In the calculation the absolute fluid pessue is taken as the atmospheic pessue. It is clea fom Fig. 2 that at a specific degee of satuation the value of B is stongly dependent on the atio of. Fo example, fo the poous medium with a poosity of. 5, B at full satuation is about.999 fo ¼., but only. 565 fo ¼. 5. On the othe hand, Fig. 2 implies that, fo a specific value of B, the poous medium with a stiffe solid skeleton can achieve a highe degee of satuation. Figue 3 illustates the effects of the paticle compessibility and of the poosity on the P-wave velocity in fully satuated poous media, whee the solid lines ae geneated using equation (2) (i.e. the paticle compessibility is taken into account) and the boken lines ae geneated using equation (22) (i.e. the paticle compessibility is neglected). In computation the shea modulus, ì, is elated to using a 2 Solid line: Autho Boken line: Skempton 2 Solid line: Autho Boken line: Kokusho Poosity, n 5 5 Poe pessue coefficient, B Poosity, n 5 5 n 5 5 Poe pessue coefficient, B n n 5 35 Bulk modulus atio, n 5 35 Bulk modulus atio, Fig.. Influence of paticle compessibility on poe pessue coefficient

5 254 YANG 5 5 Poe pessue coefficient, B 5 Poe pessue coefficient, B Degee of satuation, S Degee of satuation, S Fig. 2. Poe pessue coefficient as a function of degee of satuation: poosity, n.35; poosity, n Poosity, n Poosity, n 5 35 P-wave velocity, V p : m/s n 5 35 n 5 5 P-wave velocity, V p : m/s Poosity, n 5 5 Poosity, n Bulk modulus atio, Bulk modulus atio, Fig. 3. Influence of paticle compessibility on V p (solid line, paticle compessibility included; boken line, paticle compessibility excluded): S %; S 95% common value of Poisson s atio,.3, and the specific gavity of solid gains is assumed as It is evident fom Fig. 3 that fo poous media with small poosity the diffeence between the values of V p given by equations (2) and (22) is vey significant and exists ove an entie ange of the atio. Fo poous media of high poosity, howeve, the diffeence between the pedictions given by equations (2) and (22) becomes less significant, especially at low values of. In the case of patially satuated poous media with eithe low o high poosity, the deviation of the value of V p due to neglect of the paticle compessibility is too slight to distinguish, as shown in Fig. 3. The deived explicit expession in equation (25) enables a quantitative identification of the effect of paticle compessibility on the elationship between B and V p, as shown in Fig. 4, whee the thick lines ae obtained with equation (25) and the thin lines ae geneated by equation (5), a limiting case

6 6 PORE PRESSURE COEFFICIENT FOR SOIL AND ROCK P-wave velocity, V p : m/s P-wave velocity, V p : m/s Poe pessue coefficient, B Poe pessue coefficient, B Fig. 4. Relation between B and V p (thick line, paticle compessibility included; thin line, paticle compessibility excluded): poosity, n.5; poosity, n.35 Poosity ¼. 35 Table. Values of B and V p at full satuation fo vaious atios of Bulk modulus Poosity ¼. 5 atio, Paticle compessibility Paticle compessibility Paticle compessibility Paticle compessibility excluded included excluded included B V p : m/s B V p : m/s B V p : m/s B V p : m/s Note: The two shaded aeas indicate espectively the epesentative cases fo ock and soil. of equation (25). In the same gaph the influence of poosity and the skeleton compessibility is also illustated. Figue 4 indicates that, fo poous media of low poosity, the lage the value of, the moe significant the deviation due to neglect of the paticle compessibility. Fo instance, fo a ock with ¼. 5, the pedicted value of B at full satuation is.566 by using the geneal expession in equation (8), and the coesponding V p is aound 3753 m/s. If the compessibility of solid paticles is ignoed, the pedicted value of B becomes. 7 and the coesponding V p is as lage as 5335 m/s (Table ). Fo poous media of high poosity, the diffeence caused by ignoing the paticle compessibility will become less significant, as shown in Fig. 4. But caution should still be execised, since the diffeence may still be appeciable at high values of (Table ). CONCLUDING REMARKS With the pupose of claifying the diffeence between the expessions given by Bishop (973) and by Kokusho (2), the poe pessue paamete B was evisited in this study. It has been shown that Bishop s expession can also be deived in a igoous way based on classical Biot s theoy, and the eo in the expession by Kokusho (2) was caused mainly by the incoect use of the effective stess equation. In the same context of Biot s theoy, a fist poposal of the explicit elationship between the poe pessue coefficient, B, and the compessional wave velocity, V p, has been established. The elationship accounts fo the compessibility of solid paticles in a consistent way and is applicable to both soils and ocks. The effects of skeleton compessibility, paticle compessibility and poosity have been identified numeically with espect to B and its elation to V p. The esults indicate that majo eos in the value of B will aise fom neglect of the paticle compessibility as the skeleton compessibility appoaches the compessibility of solid gains, and meanwhile the amount of eos is sensitive to the value of poosity. Fo poous media of low poosity and low skeleton compessibility, such as ocks, the effect of paticle compessibility may be vey significant on the values of

7 256 YANG both B and V p at full satuation, wheeas fo poous media of high poosity and high skeleton compessibility, such as soils, neglect of the paticle compessibility leads only to a slightly geate value of V p but almost the same pediction fo B. ACKNOWLEDGEMENTS This wok was funded by the Reseach Gants Council of Hong Kong and the Reseach Committee of The Univesity of Hong Kong unde Gant No NOTATION B poe pessue coefficient g acceleation of gavity bulk modulus of solid skeleton K f bulk modulus of poe fluid K s bulk modulus of solid gains K w bulk modulus of poe wate k9 pemeability (m 2 ) k pemeability coefficient (m/s) n poosity p f poe pessue S degee of satuation u i displacement of solid phase V p compessional wave velocity V s shea wave velocity w i elative displacement of fluid phase Æ, M paametes accounting fo the compessibility of constituents å ij stain tenso ç fluid viscosity º Lamé constant of solid skeleton ì shea modulus of solid skeleton total density f density of fluid density of solid gains s ó ij ó 9 ij total stess effective stess REFERENCES Biot, M. A. (956). Theoy of popagation of elastic waves in a fluid satuated poous solid. J. Acoust. Soc. Am. 28, Biot, M. A. (962). Mechanics of defomation and acoustic popagation in poous media. J. Appl. Phys. 33, Bishop, A. W. (973). The influence of an undained change in stess on the poe pessue in poous media of low compessibility. Géotechnique 23, No. 3, Ishihaa, K., Tsuchiya, H., Huang, Y. & Kamada, K. (2). Recent studies on liquefaction esistance of sand-effect of satuation. Poc. 4th Int. Conf. Soil Dynamics Geotech. Eathquake Engng, San Diego. Kokusho, T. (2). Coelation of poe pessue B-value with P-wave velocity and Poisson s atio fo impefectly satuated sand o gavel. Soils Found. 4, No. 4, Skempton, A. W. (954). The poe pessue coefficients A and B. Géotechnique 4, No. 4, Yang, J. (22). Liquefaction esistance of sand in elation to P-wave velocity. Géotechnique 52, No. 4, Yang, J. & Sato, T. (998). On the vaiation of Poisson s atio in shallow soil layes. Poc. 53d Nat. Conf. of Japan Society of Civil Enginees, Kobe, Japan, III-A395, Yang, J. & Sato, T. (2a). Intepetation of seismic vetical amplification obseved at an aay site. Bull. Seismol. Soc. Am. 9, No. 2, Yang, J. & Sato, T. (2b). Computation of individual contibutions of two compessional waves in vibation of wate-satuated soils. Comput. Geotech. 27, 79. Zienkiewicz, O. C. & Bettess, P. (982). Soils and othe satuated media unde tansient, dynamic conditions: geneal fomulation and the validity of vaious simplifying assumptions. In Soil mechanics: Tansient and cyclic loads (eds O. C. Zienkiewicz and G. N. Pande), pp. 6. Chicheste: John Wiley & Sons.