An introduction to continuum mechanics and elastic wave propagation

Transcription

1 An ntroducton to contnuum mechancs and elastc wave propagaton lecture notes Authors: Mhály Dobróka, professor Judt Somogyné Molnár, research fellow Edtor: Judt Somogyné Molnár, research fellow Department of Geophyscs Unversty of Mskolc October 4

2 Contents Introducton 3 Contnuum mechancal overvew 3 Deformatons and strans 4 The moton equaton 3 Materal equatons 5 3 Materal equaton of the perfectly elastc body, stress dependent elastc parameters 5 3 Materal and moton equaton of Hooke-body 7 33 Flud mechancal materal models and ther moton equatons 34 Rheologcal materal models and ther moton equatons 4 3 Wave propagaton n elastc and rheologcal meda 4 3 Low ampltude waves n deal flud 4 3 Low ampltude waves n sotropc lnearly elastc medum Low ampltude waves n vscous flud Low ampltude waves n Kelvn-Vogt medum 49 4 Elastc wave propagaton 57 4 Descrbng the presuure dependence of longtudnal wave velocty 57 4 The pressure dependent acoustc velocty model 58 4 Epermental settng, technque and samples 6 43 Case studes 6 4 Descrbng the pressure dependence of qualty factor 68 4 The pressure dependent sesmc q model 7 4 The pressure dependent propagaton velocty Epermental samples Estmaton of model parameters appearng n the model76 45 Inverson results 8

3 Introducton Models are the smplfed realty, where we keep the most mportant features and neglect the propertes whch do not or not substantally nfluence the eamned process In contnuum physcs the characterstcs of the materal are descrbed by contnuous functons whch s nconsstent wth the atomc structure However n the descrpton of many phenomenas (eg elastcty, flows) the atomc and molecular descrptons are not necessary The soulton s that we ntroduce the phenomenologcal descrpton method by averagng the atomc effects and take n so-called materal characterstcs "constants These characterstcs are usually non constants they are depend on temperature or other quanttes Thus we obtan a smplfed - contnuum - model of the materal whch s appled n many areas of rock mechancs and rock physcs The theory descrbng the mechancal propertes of the contnuum, the contnuum mechancs s a phenomenologcal scence Contnuum mechancal overvew Based on the contnuty hypothess densty functons assgned to etensve physcal quanttes (mass, momentum, energy) are consdered mathematcally as contnuous functons of the locaton coordnates Thus eg mass-densty functon defned as follows,, 3 m lm, () V V where V s a small volume around the pont P,,, m s the mpled mass The 3 boundary transton V s nterpereted physcally e n the equaton dm dv correspondng () the voulme dv s physcally nfntesmal The boundary transton V at low volume n mathematcal sense leads to that the materal belongs to V wll be qualtatvely dfferent The boundary transton V can be understood as V tends to a volume V whch s qute large on the atomc scale but on the macroscopc scale t s small (small enough to be consdered as pont-le) It can be seen that the contnuum mechancal and atomc descrpton methods can be compatble In contnuum mechancs we get the smplfed descrpton of large groups of atoms thus we obtan relatve smple equatons We deduce general laws hence further equatons charactersng the specfc propertes of the materal are always necessary These are the materal equatons contanng so-called materal 3

4 Mhály Dobróka 4 constants (eg elastc modul) whch reflect the neglected atomc features In contnuum mechancs the functons representng physcal quanttes are fractonally contnuous, e t may be est surfaces n the meda (eg layer boundares) along whch the respectve quanttes suffer fnte "hop" Boundary condton equatons defned along these surfaces must be met Deformatons and strans After the aom of the knematcs of deformable bodes the general movement (f t s qute small) of suffcently small volume of the deformable body can be combned by a translaton, a rotaton and an etenson or contracton took n three orthogonal drectons In the framwork of contnuum mechancs the dsplacement s gven by the contnuous vector r,t s To llustrate the meanng of the aom of knematcs let us take up the coordnate system n the pont P of the deformable contnuum and consder pont P (orgnated from the small volume assumed around P ) close to P Durng the movement of the contnuum P pass through pont ' P satsfyng the vector equaton s r r', where s s the dsplacement vector connenctng ponts P and ' P Assume that fracture surface s not etend between ponts P and ' P Then the two (adjacent) ponts can not move ndependently of one another, there s a materal relatonshp" between them defned by the contnuum materal Ths can be epressed mathematcally that the dsplacement n pont P s orgnated to characterstcs refer to pont P, or n other words the dsplacement s eerted nto seres around pont,, P : u u u u u u 3 3 u u u u u u 3 3 u u u u u u , ()

5 Contnuum mechancal overvew 5 where means other "hgher parts" appearng n the epanson and nde ndcatng net to the dervatves refers to that the dervates should take n the orgn ( P ) In the aom of knematcs we talk about the dsplacement of suffcently small volume It means that n equatons () the "hgher parts" contanng the powers and products of coordnates 3,, are neglgble, e we lve wth a lnear appromaton In addton, we also assume that the frst dervatves are small n the sense that ther product and powers are neglgble Herewth () takes the followng form,,3, u u u 3 j j j () In the followngs, we wll apply the so-called Ensten's conventon wth whch () can be wrtten as j j u u u e f an nde (or ndces) n an epresson occurs twce we should sum from to 3 In the followngs the nde besde the dervatves j u s omtted, so u u u j j (3) The dervatve tensor j u can be dvded nto symmetrc and antsymmetrc parts as u u u u u j j j j j Wth ths we obtan the followng equaton for the (3) dsplacements j j j j j j u u u u u u, (4) where u s the same for any ponts of the small volume taken around P, e u means homogeneous translaton for the movement of these ponts

6 Mhály Dobróka 6 s transzl Introducng the notaton u,u,u 3 rot s (5) t can be easly seen that the second part n (4) equals to the vectoral product [, r ] whch descrbes the rotatonal dsplacement where s rot rot s, r Thus, t s obvous that the thrd part of (4) provdes the deformaton dsplacements d d d s u,u, u, def d u Introducng the notaton 3 u u j j j (6) j u u j j j (6) can be reformulated as d u j j (7), (8) where the symmetrcal second-order tensor j called deformaton tensor To clarfy the components of the deformaton tensor take a materal lne n unt length along the coordnate as of the orgnal coordnate system and denote t as vector r,, Durng deformaton ths transforms to vector r ',, due to equaton (8), accordng to the equaton r' r s Hence the relatve epanson s ' r 3 3, (9)

7 Contnuum mechancal overvew as we lmt ourselves to small deformatons,, 33 Thus deformaton means the epanson of the secton n unt length taken along as or n other words the relatve epanson measured along as Elements, have 33 smlar meanng Elements n the man dagonal of the deformaton tensor gve the relatve epanson of the materal lne sectons fallng nto aes,, 3 To nvestgate the meanngs of the elements outsde the man dagonal let us take the unt vector j fallng nto the drecton of the coordnate as whch after deformaton transforms nto vector r ',, ' ' By equaton (9) plus formng the scalar product r, as products of deformatons one obtans Applyng ' ' r,r 3 ' ' ' ' r,r r r cos, r and neglectng the squares as well where r ' ' r and the angle between the two vectors s results as where, sn was used to small angles Ergo deformaton s the half of the angle change whch s suffered by the lne secton taken n the orgnally perpendcular drectons and j Take a prsm wth a volume V abc and wth edges parallel to the coordnate aes n the undeformed contnuum! The volume of the prsm generated durng deformaton wll be appromately ε ε ε abc V' 33 e the relatve volume change s 7

8 Mhály Dobróka V' V 33 V The sum of the elements n the man dagonal of the deformaton tensor (aka the spur of the deformaton tensor or the nvarant of the frst scalar) means the relatve volume change Contrarwse one can epresses t wth equaton qq (summarze to q!), or on the bass of defnton (7) of the deformaton tensor u u u3 dv s () 3 Ths quantty s unchanged durng coordnate transformaton To characterze the deformatons t s used to ntroduce the sphercal tensor E ( ) qq () 3 and the deformaton devatorc tensor E qq, () 3 where s a unt tensor e 8, ha k, ha k The name s orgnated n that the second-order tensor surface ordered to the deformaton sphercal tensor s sphere Wth ths tensor the pure volume change can be separated from the deformatons The remanng part of deformatons E shows the devaton from the pure volume change, e the so-called dstorton It s obvous accordng to () that E qq The decomposton of the deformaton tensor E E (3) ( ) means also the pck apart t to the volume change-free "pure dstorton" and the pure volume change The dynamc nterpretaton of the movement of the contnuum requres to ntroduce the force denstes Eperences show that the forces affectng on the contnuum can be dvded nto two types: volume and surface forces The volume force df - affectng to the

9 Contnuum mechancal overvew contnuum contans the (physcally) nfntesmal volume element dv took at a gven pont of the space - can be wrtten as df * f dv, where * f s the volumetrc force densty The ntegral of the volumetrc force densty F V f * dv gves the force affected to fnte volume The volumetrc force densty can be calculated otherwse by the defnton * F f lm V V There are forces that are physcally drectly proportonal not to the volume, but the mass These can be characterzed by the mass force densty F f lm, m m where m s the mass contaned n the volume V Wth whch or otherwse V F * f lm f V m V f * f (4) Another class of forces arsng n the contnuum are the surface forces The surface force densty can be formulated as F n lm, (5) A A where the boundary transton A can be nterpreted as A tends to a so small surface A whch s neglgble (pont-le) n macroscopc pont of vew but t s very large compared to the atomc cross secton Otherwse equaton (5) can be wrtten as F n da (6) 9

10 Mhály Dobróka In (6) nde n mples that the surface force at a gven space depends on not only the etent of the surface but ts drecton - characterzed by the normal unt vector n - too Based on (6) the force affected on the fnte surface A can be calculated as F da (7) n A Snce the unt normal vector n can pont to nfnte number of drectons apparently the knowlege of nfntely many surface force denstes s necessary to provde the surface forces However t can be proved that n n (8) n n 3 3 Ths equaton shows that f we know n one pont the surface force densty affected on three orthogonal coordnate plane then surface force denstes (also known as strans) affected on any n drectonal surface can be calculated by the help of equaton (8) Introducng the notatons n n, n, n3,, 3,, 3 3, 3, 3 (8) can be wrtten as 33,,,3 (9) n j n j (where accordng to our agreement one has to summarze to j from to 3) Ergo after (9) to characterze the surface force densty the second-order tensor j s ntroduced whch s the j-th component of the stran vector affected on the surface suppled wth a normal pontng to the drecton of the -th coordnate as It can be proved that ths tensor s symmetrc, e j j The elements,, 33 n the man dagonal of the tensor are normal drectonal (tensle or compressve) stresses, the outsde elements, 3, 3 are tangental (shear or slp)

11 Contnuum mechancal overvew stresses Smlarly at the deformaton tensor one can produce the stress tensor as the sum of the devator and the stress sphercal tensors where and T T, () T qq, T qq, () 3 3 qq s the sum of the elements n the man dagonal of the stress tensor The moton equaton At the deducton of the moton equaton of deformable contnua the startng pont s Newton's II law whch says that the tme-dervate of the mpulse of the body equals to the sum of the arose forces di F dt The mpulse of the body can be determned by the formula I dv V where v m lm v V V s s the volumetrc mpulse densty, whle v s the velocty The resultant force affectng t the body s the sum of the volume and surface forces, e F f dv da V A The ntegral form of the moton equaton can be wrtten as d v dv dt V t n f dv n da, t At V where V ( t ), A ( t ) are the volume as well as surface movng together wth the contnuum To the -th coordnate of the vector equaton one can obtan

12 d dt V Mhály Dobróka v dv f dv n da () t V t At To transform the latter equaton use the dentty d dv dv v dv dt V t V t t and the Gauss-Osztogradszkj thess n da j n j da j da j da dv dv At At At At V t, where daj n j da and,, 3 denotes - as a formal vector - the -th row of the stress tensor Here the equaton (9) was also used Now the moton equaton () can be wrtten as v V t t dv v v f dv dv Hence volume V ( t ) s arbtrary, from the dsappearance of the ntegral one can nfer to the dsappearance of the ntegrand v t dv v v f dv, otherwse v t dv v v f k () Ths s the local form of the moton equaton of the deformable contnuum, also known as the balance equaton of the mpulse In contnuum theory equatons descrbng the transport of etensve quanttes can be commonly reformulated to the format of the contnuty equaton If the bulk densty of a quantty s w then the convectve current densty of the gven quantty s denoted as J konv w wv kond J represents the conductve (connected to macroscopc motons) current densty Then w the balance equaton of quantty w s

13 Contnuum mechancal overvew w dv t or f t has sources (or snks) w dv t konv kond J J, w konv kond J J, w w w (3) where s the source strength whch provdes the quantty of w produced or absorbed n unt volume per unt tme Introducng the convectve J konv mp v v and conductve J mpulse-current densty vectors equaton () can be wrtten as v konv kond t dv J mp J mp f kond mp Ergo the stress tensor (that of onefold) s physcally the conductve mpulse current densty, whle volumetrc force densty f plays the role of the source strength of the mpulse It s well-known that smlar balance equaton can be formed to (mass) densty component of a flud compound j of the j-th j konv kond dv J m J m t j where the convectve mass flow densty s, J konv m j v The conductve mass flow densty provdes a way to decrbe the dffusve motons, source strength j refers to the chemcal reacton whch gves the producton of the j-th component To one-component flud by neglectng the source strength the contnuty equaton dv t v (4) provdes the balance equaton of the mass By transformng the left sde of equaton () one can obtan v dv t v v t v grad v f k, where the dentty dv a A adva Agrad ( a ) 3

14 Mhály Dobróka was used (where a and A are the contnuous functon of the three spatal coordnates) By takng nto consderaton the contnuty equaton, () can be wrtten as v t v grad v f k We call the partal dervate dervate and otherwse local, whle the operator v grad t s the convectve d v grad dt t s the substantal dervatve Hereby the moton equaton can be wrtten as dv f (5) dt k In sold contnua the convecton can be neglgble thus d and the moton equaton s dt t In vector form u f (6) t k v t v grad v f Dv corresponds wth the moton equaton (5), whle s f Dv t, (7) (8) refers to equaton (6), where Dv s the sgn of the tensor dvergence and the double underlne denotes the tensor The contnuty equaton (4) and equaton (5) are the contnuum mechancal formulaton of the mass conservaton and the mpulse thess respectvely, e epress general (vald for any contnuum) law of nature However there are scalar unknowns n these four scalar equatons (assumng the mass forces f as knowns) Thus equatons derved from natural basc law are sgnfcantly underdetermned, so unambgous soluton can not be ested 4

15 Contnuum mechancal overvew To clearly descrbe dynamcally the movement of the contnuum more s equatons are requred whch can be obtaned on the bass of restrctve condtons took to the materal qualty of contnuum and ts elastc propertes These equatons are the materal equatons wrote to the s ndependent elements of the stress tensor 3 Materal equatons Elastc propertes of materal contnua are very dverse A general materal equaton whch comprse all of ths varety, does not est Instead, one should hghlght from all elastc propertes of the nvestgated medum the most relevant ones and neglect the other "dsturbng" crcumstances Ths can be epressed dfferently e we create a model The most mportant smple and comple materal models bult from the smple ones wll be descrbed n the followngs especally consderng the rock mechancs and sesmc/acoustc aspects 3 Materal equaton of the perfectly elastc body, stress dependent elastc parameters Perfectly elastc body means that stresses depend on the deformatons domnant at a gven space of the contnuum n a gven tme, e The functon f,, 33,, 3, 3 (3) f s generally non-lnear However, very often we deal wth small stress change related to small deformatons For eample, f an elastc wave propagates n a medum ested n a gven stress state, the wave-nduced deformaton and stress perturbaton s very small compared to the characterstcs of the orgnal, statc load of the medum In ths case, the functon where the notatons f can be appromated by the lnear parts of ts power seres 6 f, (3),, 33,, 3,,, 33,, 3, 3 3 5

16 were ntroduced The constants c 6 Mhály Dobróka f are named elastc constants whch character- ze the perfectly elastc body near the undeformed state Obvously, the seres epanson can be performed around any deformaton state, when Then, or because of f 6 f f, f 6 (33) Small deformatons superposed to the basc (or equlbrum) deformaton are -smlary to (3) - connected to stress change by the equaton (33), but elastc constants c f depend on the basc deformatons If the functon f can be nverted, e g ( ) the elasc propertes c ( ) depend on stress state Ths s supported by the sesmc eperence that the velocty of elastc waves s a functon of n-stu stress state Snce the propagaton velocty depends on elastc modul one can see that the model of the perfectly elastc body can be phenomenologcal sutable to descrbe the velocty/pressure relatonshp through the pressure-dependent modul Of course the producton of the approprate materals equaton depends on rock type and rock qualty Snce waves mean small deformaton, ths tme the seres epanson (33) provdes good appromaton Elastc parameters c f n (33) form a 66 matr It can be

17 Contnuum mechancal overvew proved by the help of the energy thess formulated to contnua, ths matr s symmetrc Ths means that n general case, the elastc propertes of the ansotropc contnuum can be characterzed by ndependent elastc parameters The propertes of materal symmetry can sgnfcantly reduce the number of elastc constants Isotropc contnuum can be characterzed by two elastc constants In case of several practcal nstance, lnear appromaton (33) denotes a good appromaton but n sesmcs apart from eploson ssues That materal model n whch (33) s vald for not only small deformatons, s called the model of lnearly elastc body In small deformaton nterval the model of perfectly elastc body transforms to lnearly elastc body 3 Materal and moton equaton of Hooke-body The phenomenologcal descrpton of ansortopy s very mportant n rock physcs and sesmc too However, the smplfcaton s reasonable n sesmc practce the most wdely used lnearly elastc medum model assumes sotropy The lnearly elastc sotropc body s characterzed by only two elastc parameters whch can be ntroduced a number of ways Wth thermodynamc consderatons the two parameters are the so-called frst and second Lame coeffcents wth whch the materal equaton of the lnearly elastc sotropc body or Hookebody can be wrtten as (34) Hence to the man dagonal of the stress tensor one can obtan the equaton where qq 3 K, (35) K s the compresson modulus 3 Introducng the stress sphercal tensor T qq, 3 on the bass of (35) ts relatonshp wth the deformaton sphercal tensor s T 3K E (36) The stress devatorc tensor T E T after (34) s T (37) 7

18 8 Mhály Dobróka The materal characterstc parameters and are generally depend on temperature too In engneer lfe nstead of Lame coeffcents the Young's modulus E and Posson's number m are often used In case of unaal load (eg a thn long rod clamped at one end, ts other end s pulled) f s aal, the tensle s E, (38) so the Young's modulus E can be determned drectly In the plane perpendcular to the tenson the deformatons, 33 have opposte sgn and proportonal to the relatve epanson m 33, where m s the Posson's number The relatve volume change s m 33 (39) m Snce n case of (stretchng) the volume can not decrease so from (39) m The equalty refers to ncompressble materals (eg statc load n flud) To look for the relatonshp of parameters (39) to (34) At unaal load, and E, m wrte the epresson of m, (3) m on the other hand because of qq (snce there s only one stress component ests) m 3 (3) m By comparng the equatons (3) and (3), as well as (38) or m, E 3

19 Contnuum mechancal overvew m m E, m m m E One can obtan the moton equaton of the lnearly elastc sotropc body f one substtutes the materal equaton (34) to the general moton equaton (6) By formng the dvergence of the stress tensor (34) n case of homogeneous medum ( and are ndependent from locaton) k u k k u k k, k where the defnton (7) of the deformaton tensor was used and one must sum for the same ndees Snce u k k u k k and k, the moton equaton can be wrtten as u f u, (3) t n vectoral form by usng () s f s t grad dv s (33) Ths equaton alas the Lame equaton s the moton equaton of the lnearly elastc homogeneous body (Hooke-body) Mathematcally (33) s an nhomogeneous, second-order nonlnear coupled partal dfferental equaton system To obtan ts unambguous soluton ntal and boundary condtons are necessary Settng the ntal value problem means that we requre s r, v r, at each pont of the tested V volume n t the dsplacement and velocty 9

20 Mhály Dobróka Boundary condtons requre the dsplacement s r *,t and the value of the drectonal (normal) dervatve s n at any t tme n * r ponts of the surface A boundng volume V In case of nhomogeneous lnearly elastc sotropc body the Lame coeffcents depend on space:,,,,, be wrtten n the form So the dvergence of the stress tensor (34) can 3 3 k u u u k dv s dv s k k k Wth whch the moton equaton s u t f u u u k dv s dv s k k k or n vectoral form s f s grad dv s t where grad,rot s denotes vectoral multplcaton grad grad s grad,rot s grad dv s 33 Flud mechancal materal models and ther moton equatons Fenomenolgcal defnton of fluds s based on the eperence that the smaller the tangental (shear) stresses occurrng n fluds are the slower the deformaton s By etrapolatng ths observaton we consder that contnuum as flud, n whch shear stresses do not occur n repose state, e elements outsde the man dagonal of the stress tensor are dsappeared n every coordnate system In sotropc fluds, elements n the man dagonal are equal, e the stress tensor n repose state s, p where p s the scalar pressure Materal and moton equaton of deal flud (Pascal s body) We call that flud deal n whch shear stresses durng moton do not occur, e the stress tensor of the deal flud for any deformaton s

21 Contnuum mechancal overvew (34) p Snce then T, the stress tensor of the deal flud s a sphercal tensor Ths s another formulaton of the well known - from flud mechancs - Pascal's law, therefore the deal flud called otherwse Pascal's body Equaton (34) only makes constrant to the format of the stress tensor, but t s not a materal equaton The materal equaton usually connects the stresses wth knematc characterstcs In contrary n flud mechancs the pressure s nvestgated n densty and temperature dependence For eample f pressure depends on only densty p p, we talk albout barotrpoc flud Equaton (34) s vald n case of gases too The state equaton of deal gases can be wrtten as p T R, where R s the gas constant and T s the absoulte temperature The state equaton s smpler n case of specal change of state Eg at sothermal processes p konstans, whle n case of adabatc change of state p konstans, c p where, c p s the specfc heat mesured at constant pressure as well as c v s at constant c v volume, respectvely Based on (5) and (34) the moton equaton of the deal flud s v t p v grad v f or n vectoral form

22 Mhály Dobróka v v grad v f grad p (35) t Ths equaton s the so-called Euler-equaton Materal and moton equaton of the Newtonan flud The deal flud model not enables to descrbe a number of practcal problems It s a general eperence that waves absorb n fluds or frcton losses occur n fluds durng flowng To eplan these phenomena an mproved flud model s requred At the phenomenologcal defnton of fluds we hghlghted that shear stresses are the smaller the slower the deformaton s Ths means that stresses n flud orgnated from frcton depend on the swftness of the deformatons, the deformaton velocty tensor t v vk k e ' f,, 33,, 3, 3 In terms of geophyscal applcatons only the sotropc fluds have sgnfcance whch show lnear depencence n deformaton veloctes Then (because of the sotropy) wrtng the tensor nstead of deformatons materal equaton ' n the (34) formula of the tensor, one obtans the, (36) where and are the vscous modul Ths s the materal equaton of the Newtonan fluds (Newton's body) Introducng the deformaton velocty sphercal tensor E 3 and the deformaton velocty devator tensor E 3 equaton (36) can be dvded nto two tensor equatons

23 Contnuum mechancal overvew T E,T 3 v E, (37) where v s the so-called bulk vscosty 3 In realty, to descrbe the frctonal fluds the materal equatons of the Pascal s and Newton's body should be combned, e the total stress tensor s p By formng the dvergence of the tensor and usng (36) one obtans k p v k k v, wth whch the moton equaton (5) can be wrtten as v p v grad v f v dv v, (38) t or n vectoral form v v grad t v f grad p v grad dv v Ths s the moton equaton of frctonal fluds, e the Naver-Stokes equaton The Naver-Stokes flud Eperences denote that (relatve to the sound velocty), at low-velocty flows and low frequency sound waves the bulk vscosty n (37) can be consdered appromately zero (The measurement of v s dffcult because of ths small effect whch becomes possble prmarly n case of hgh-frequency ultrasound eperments) Therefore the Newton-model can be constrcted n sesmc and rock mechancal applcatons It allows us to create a new flud model n whch because of v, (39) 3 therefore the stress tensor nstead of (36) s ', (3) 3 3

24 and due to (3) equaton (37) s Mhály Dobróka T E, T (3) Equatons (3) or (3) are the materal equaton of the so-called Naver-Stokes s body Due to (39) the moton equaton (38) can be wrtten as v p v grad v f v dv v, t 3 or n vectoral form v t v grad v f grad p v grad dv v 3 34 Rheologcal materal models and ther moton equatons The materal equatons of Hooke s body (34) and Newton s body (36) n elastc aspect descrbes two mportant lmt cases of sotropc materal contnua: the lmt case of stresses depends on only the deformatons (34), as well as depends on (lnearly) only the deformaton velocty (36) respectvely In realty, the stress tensor of the medum (n more or less scale) depends on the deformatons and the deformaton veloctes f,, or otherwse the materal equaton can be wrtten n the general form of F,, (3) In many cases, ths materal equaton can be produced accordng to the equatons (34) and (36) or (34) and (3), n other words the materal model descrbng the elastc propertes of the medum can be bult from Hooke s and Newton's body or Naver-Stokes s body In ths case we are talkng about a comple materal model Often, the stress change velocty tensor plays a role n the materal equaton, e the materal equaton s F,,, (33) 4

25 Contnuum mechancal overvew The functon F s usually lnear appearng n (3), (33), the so-called rheologcal equatons Then the stress tensor s the lnear epresson of tensors and to whch one wll see some eamples n the followngs, The materal and moton equatons of the Kelvn-Vogt s body The Kelvn-Vogt model shows the smplest combnaton of the Hooke s and Newton s body Fgure llustrates the model Fgure : Model of the Kelvn-Vogt s body The sprng models the Hooke's body, whle the perforated pston movng n the vscous flud-flled cylnder models the Newton's body In case of one-dmensonal motons t s obvous that the dsplacements on the two body parts are equal, whle the sum of the forces arose n the two branches of the model are equal to the forces affected to the model Ths smple crteron can be generalzed as follows H N (34) H N (35) By usng the materal equatons (34) and (37), (34), (35) provdes the followng result (36) 5

26 Mhály Dobróka whch s the materal equaton of the Kelvn-Vogt s body Based on (36) to the stress devator tensor one obtans T E E (37) or by ntroducng the so-called retardaton tme one gets the equaton T (38) t E One can see that n case of slow processes ths materal equaton pass through the materal equaton of the Hokke s body If t s the characterstc tme of the process then E t the order of magntude of the dervatve In case of slow processes t gves then ndeed T E In case of fast processes, t then the equaton (38) can be appromated by T E whch s the devator equaton of the Newton s body due to The equaton of the spercal tensors are T ( ) 3K E 3 v E, (39) where K, v 3 3 If the Naver-Stokes s body descrbes the vscous forces nstead of Newton s body, the devator equaton remans unchanged but for the equaton of the sphercal tensor nstead of (39) one obtans 6

27 Contnuum mechancal overvew T 3K E (33) Ths appromaton s adequate for the descrpton of rock mechancal and sesmc phenomena n many cases Solve the dfferental equaton (38) to analyze the propertes of the Kelvn-Vogt s body Introducng the notaton G T E to (38) one gets the equaton G T (33) G We look for the soluton by the method of varyng constants n the followng form G t t c e (33) To the functon c from (33) one obtans the equaton c e t T t from where c t t' e T t' dt' K, where K s constant Thus based on (33) the soluton of (33) s T t t t' E e K e T ( t' ) dt' (333) The ntal condton n t s specfed n the form of T T ( ),E, therefore K T ( ), so from (333) one gets the equaton E T T ( ) e t t t t' e T dt (334) 7

28 One can see that the deformatons Mhály Dobróka E are dffer from the value T accordng to the Hooke s body and show eplct tme dependence If eg we consder that specal case when we load the Kelvn-Vogt s body by constant stress T, then due to T T ( ) t E T ( ) e, e deformatons appromate asymptotcally the value E T ( ) got based on the Hooke s model Parameter clarfes the velocty of ths appromaton Ths s that tme durng whch E puts the -fold of the asymptotc value on e Snce the deformaton of the Kelvn-Vogt s body reaches the value belongs to the Hooke s body only delayed (retarded), the rheologcal parameter s called retardaton tme The rock mechancal process descrbed above and llustrated n Fgure s called creep Fgure : The phenomenon of creep and the geometrc meanng of parameter Returnng to the general equaton (334), by partal ntegraton one obtans from t an ntal condton ndependent formula E e t tt' T t' dt' 8

29 Contnuum mechancal overvew Herewth contrastng the Kelvn-Vogt s body wth the Hooke's body - an eample can be seen to that the deformatons n the body n gven tme t depend on not the value of the stresses n the same tme but the stesses took n the prevous nterval,t One obtans the moton equaton of the Kelvn-Vogt s body by substtutng the materal equaton (36) to (6) or n vectoral form u f u dv s v dv v, (335) t s f s t grad dv s v grad dv v (336) The materal equaton of the Mawell body As t was presented, the Kelvn-Vogt body bult up from the Hooke and Newton bodes behaves as lnearly elastc body n case of statc border-lne case, and as vscous flud n case of fast processes Another materal model can be bult up from the Hooke and Newton bodes as well, whch acts le flud n slow processes and as elastc sold contnuum n fast ones Ths s the model of the Mawell body llustrated sematcally n Fgure 3 Fgure 3: The model of the Mawell body Thnkng about the one dmensonal motons based on the fgure t can be seen that the same forces arse n the two elements of the model and the sum of the dsplacements of the two elements s equal to the total dsplacement Generalzed ths, the eqatons 9

30 Mhály Dobróka H N H N (337) can be used as basc equatons n the deducton of the model s materal equaton Based on the materal equatons of the Hooke and Newton bodes and H H (338) N N (339) H From (338) the H qq 3, or Wth ths the equaton 3 qq H 3 qq can be obtaned for the (337) H deformatons smlarly from (338) However accordng to 3 N qq, 3 N qq 3 and so the materal equaton of the Mawell body can be wrtten based on (339) as 3 qq (34) For the trace of the tensor the 3 qq qq 3 3 equaton, and so for the sphere tensors the T T 3 o v E (34)

31 Contnuum mechancal overvew equaton arse, where 3 v 3 K s the volumetrcal relaaton tme Consttutng the dfference of (34) and (34) the devator equaton can be deduced T T E, (34) where s the relaaton tme Note that f we use Naver-Stokes body nstead of the Newton body n the Mawell model, then because of v from (34) 3 materal equaton of the Mawell body can be wrtten as, e T, hence the (343) 3 Ths appromaton can often be appled n case of rocks For slow processes ( t, t s the characterstc tme of the process) the T dervatve can be neglected n the equaton (34) Then we get the T T E appromate equaton In case of slow processes the Mawell body changes to the model of Newtonan flud In case of fast processes ( t ) T T n equaton (34), so the equaton leads to the T E or T E acts n ths border-lne case as a Hooke body Lookng for the soluton of equaton (34) n the form of for the T c t e t materal equaton It means that the Mawell body c t functon from (34) the followng result can be obtaned c e t t' E t' dt' whereby 3

32 Mhály Dobróka T e t tt' E t' dt' It can be seen from the equaton settng the Mawell body aganst the Newton body that by the Mawell body the stresses at a gven t tme depend not only on the deformaton veloctes domnatng at t, but all the of T E of prevous tmes n the,t nterval nfluence the value A typcal property of the Mawell body can be shown f the soluton relatng to tme statonary deformatons of equaton (34) s derved There the T T equaton gves the result T T e t (here the upper case does not ndcate the sphere tensor, but the value taken at t!) The eponental loss of stresses s shown n Fgure 4 Fgure 4: The phenomenon of stress relaaton, the geometrcal meanng of the parameter Ths phenomenon common n rocks s the release or relaaton of stresses The relaaton tme s the tme durng the stresses decrease to the e-th part of the ntal T value The Mawell model s bascly a flud model, no stresses arse n t aganst statc deformatons So t can be used only for eplanaton of dynamc features durng descrbng rocks 3

33 Contnuum mechancal overvew The materal equaton of the Poyntng-Thomson body The prevously ntroduced Hooke, Kelvn-Vogt and Mawell models each took hold of one mportant sde of elastc-rheologcal features of rocks: the Hooke body the resstance aganst the statc deformatons, the Kelvn-Vogt body the creepng, the Mawell body the stress relaaton The Poyntng-Thomson model or standard body s a rock mechancal model whch combnes the Hooke and Mawell bodes as t s shown n Fgure 5 and t can descrbe the three phenomena smultaneously The base equatons are H M, H M, (344) M M where are the deformaton and stress arsng n the Mawell body, Fgure 5: The model of the Poyntng-Thomson body The materal equaton of the standard body can be obtaned after addng the equatons (34) and (34) together wth markng the materal propertes of the Mawell body wth ' n (34) ' ' ' ' ' ' ' qq ' ' 3 ' ' Regardng the equaton as sum of a sphere and a devator tensor the devator tensor can be derved as 33

34 Mhály Dobróka T ' ' ' E ' E T, (345) whle the eqaton of the sphere tensor has the form of where K' T 3 K E 3 ' ' K ' E 3 K' ' ' 3 ' 3 T, (346) ' ' 3 If we use Naver-Stokes body nstead of a Newton body n the Poyntng-Thomson model v ' ', equaton (346) has the more smple form 3 T 3K E Ths assumpton s wdely used durng the descrpton of many rock mechancal processes Introducng the notatons ' ' ', ' ' ' ' 3, ' ' 3 ' ' K' 3 K ' ' 3 the eqatons (345), (346) can be wrtten n the form T E (347) t t 3 K E T (348) t t The and quanttes are called the devatorc relaaton and retardaton tmes respectvely, the and quanttes are called volumetrc relaaton or rather retardaton tme It can be seen, that n the model or rather relatons are vald 34

35 Contnuum mechancal overvew The (347)-(348) equaton system can be substtuted wth dfferent equatons dependng on the magntude of the,, and rheologcal parameters Ther scope of valdty can be gven easly by markng the typcal duraton of rock movements wth t In case of phenomena varyng very slowly n tme, e t, or rather t (347), (348) change to the materal equaton of lnearly elastc body T E, T 3 K E In case of processes varyng faster n tme, assumng the, or rather the relatons phenomena can be dstngushed, where t and t Then the equatons found vald wth a good appromaton for practcal rock mechancal processes can be wrtten (Asszony and Rchter 975) T t t T 3K E E In case of eamnaton of more faster processes, t the (347) changes agan to the materal equaton of lnerly elastc body T or n another form T E E ' Dependng on the relaton of rheologcal parameters the connecton between the sphere tensors can be: a) n case of,, f t the T 3K E eq can be obtaned b) f, and t, or rather t then besde the lnear relatonshp between the devatorc tensor, the rheologcal equaton 35

36 T t Mhály Dobróka t 3K E can be wrtten between the sphere tensors c) f, or rather s n the order of magntude of or rather, or the process s so fast that the relaton t s fulflled, then T 3 K E, or n other form T 3 K K' E, e both n the devator and n the sphere tensors lnear, but wth greater elastc modul ', or rather K K' compared to the, K modul manfested n slow (quasstatc) processes To wrte the moton equaton of a body followng the materal equaton (347)-(348), let us solve the equatons for T T, or rather T E E E T! The equaton (347) can be wrtten as Ths nhomogeneous equaton can be solved n method Lookng for the soluton n the form of T E wth the varaton of constants for the T E c c t coeffcent the c t t equaton can be obtaned, where e t t t' e E dt' K (347) can be wrtten n the followng form K denote constants Usng ths the soluton of the equaton 36

37 Contnuum mechancal overvew T t tt' E e E dt' K e (349) Based on the equaton t can be ponted out that the stresses arsng n a gven tme n the Poyntng-Thomson body depend on all the t E values n the,t nterval As we saw the Poyntng-Thomson body s a Hooke body characterzed by a ' Lamé constant n the case of fast processes Let us assume that the body s loaded very fast wth T ' E stress (here the upper case does not ndcate the sphere tensor, but the value taken at t!) Eamng a process startng from ths ntal state at t ' E E K, from whch K ' E If the deformatons do not vary n the followng e E, from (349) T ' e t E As t can be seen n Fgure 6 the stresses decrease from the ntal the ' value to E value Ths s the relaaton phenomenon n case of the Poyntng-Thomson body E Fgure 6: Stress relaaton n case of the Poyntng-Thomson body If we assume that the eamned rock mechancal phenomena proceed from the state of permanent equlbrum, we should specfy as ntal condton that the rock follows the T E 37

38 Mhály Dobróka equaton of a lnearly elastc body at t, e K In case of such phenomena the general soluton of the equaton (347) s T E e E dt' t t t' Snce the eq (348) corresponds wth eq (347) structurally, ts soluton can be wrtten drectly: t t t' T 3 K E 3 K e E dt' Equatons (347), (348) can be solved for the deformatons n a smlar manner E T e t A t t' e T dt (35) E 3 K t t t' T e B e T dt, (35) where A and B are the constants determned by the ntal condtons If the body s loaded vary fast by a the formula E T stress the arose deformatons can be derved by T, snce the standard body appromates the Hooke body n ths ' process For the processes startng from ths state the relatonshp ' A T s vald ' based on the ntal condtons If the stresses do not vary later t T can be wrtten as ' E e ' T the equaton (35) 38

39 Contnuum mechancal overvew Fgure 7: The phenomenon of creepng n the case of the Poyntng-Thomson body As t can be seen n Fgure 7 the formula descrbes the ncrease of deformatons from the T ' value to the T value Ths phenomenon s called creepng The Poyntng-Thom- son body can descrbe both the relaaton and creepng phenomena 39

40 3 Wave propagaton n elastc and rheologcal meda The objectve of geophyscal nvestgatons s mostly the determnaton of subsurface structures of the Earth - as materal half space - by usng surface measurements For some measurement methods (gravty, magnetc, geoelectrc) the mpact measured on the surface s ntegrated n the sence that the quantty measured n a gven pont reflects theoretcally the effect of the whole half space but at least a space porton to a certan depth It makes the nterpretaton much easer f the measured effect yelds nformaton from a local area of a determned curve and not from the whole half space Ths gves the smplcty of the analyss of rocks by elastc waves and ts mportance as well, because the laws of beam optc can be used for the wave propagaton n a certan appromaton In the followngs the most mportant features of elastc waves are revewed wth respect to the major materal equatons dscussed prevously We deal only wth low ampltude waves n our nvestgatons It means that the basc equatons are solved wth a lnear appromaton There s a substantal dervate on the left sde of the (5) general moton equaton, where where dv dt v grad v v v grad v, t convectve dervatve means namely a nonlnear term Its neglect requres the fulflment of a smple crtera n case of waves If T s the perodc tme of the wave, s the wavelength, A s the ampltude, then the order of magntude s u v t A T v, t A T v grad v A T A A T T The convectve dervatve can be neglected besde the v t local dervatve f v A A v grad v, e, e A If ths crtera s fulflled, then we can speak t T T about (compared to the wavelength) low ampltude waves In ths case we can wrte the lnear v t dv dervate nstead of dt 4

41 Wave propagaton n elastc and rheologcal meda The assumpton of homogeneous medum especally f t has an nfnte dmenson s unsubstantated n geophyscal aspect We stll use ths appromaton, because the most mportant propertes of the wave space, the connecton between the parameters characterzng the waves can be ntroduced most easly n case of wave propagaton n nfnte homogeneous medum We do not have to consder boundary condtons durng solvng the dfferencal equatons n nfnte homogeneous space It s a sgnfcant smplfcaton The so evolvng waves are called body waves (The assumpton of nfnte spreadng s abstracton of course, whch means the restrcton that the surfaces - maybe estng n the medum - are very far from each other regardng to the wavelength) In the followngs the propertes of body waves propagatng n nfnte homogeneous medum followng dfferent materal equatons are summerzed 3 Low ampltude waves n deal flud The moton equaton of deal flud s gven by (35) The low ampltude wave soluton of the equaton can be wrtten n the followng form v f grad p (3) t From the pont of vew of wave theory the mportance of f mass forces s confned to the determnaton of equlbrum, p dstrbutons In equlbrum the f grad p statcs base equaton s vald For eample n case of ar ths equaton determnes the densty and pressure dstrbutons n the atmosphere of the Earth Ths dstrbuton s nhomogeneous, but the nhomogenety occurs on a very large scale compared to the wave length (for eample the wave length of a Hz frequency sound has the order of magntude of m, whch s really small compared to the km order of magntude of the characterstc changes of the atmosphere) So the medum s locally homogeneous from the pont of vew of wave propagaton, e the equaton (3) can be solved for homogeneous space If the wave propagates through dstances characterzed by nhomogenety, the changes n accordance wth the place of local features (local propagaton velocty) should take nto account As the f mass force feld has no nfluence on the wave soluton n the order of magntude of wave length, we can apply the f substtuton n (3), e 4

42 Mhály Dobróka v grad p t To solve the equaton system there s a need for two more equatons, for the dv t v contnuty equaton and for a materal equaton, for eample the p p barotropc equaton of state Assumng that the wave causes the small features, e p', ' changes of the p, equlbrum p' p, ' the equaton system can be lnearzed Neglected the product of the dervatves the followng equatons can be obtaned v grad p' t p', ', v quanttes or ther (3) ' dv v t p' c ', h p where ch From the last two equatons p' v dv, (33) ch t t the dvergence of (3) can be wrtten as v dv dv grad p' p' t If we compare ths equaton aganst (33) the ' c p' p t h wave equaton can be derved The followng equaton can be smlarly deduced as well 4 v v c t h

43 Wave propagaton n elastc and rheologcal meda The monochromatc plane wave soluton of the equatons accordng to t kr ˆ e can be wrtten n the form of p' p v v * e * e t t ker ker, (34) These functons satsfy the wave equaton, but the queston s: are they the soluton of the moton equaton? As we get the equaton rot v after formng the rotaton of eq (3), t can be seen that the moton equaton s fulflled only n case of v e e the dsplacement of the wave or rather the velocty of the dsplacement s parallel to the drecton of wave propagaton The (34) functon descrbes a longtudnal wave propagatng wth c h p flud (Pascal body) s the sound wave velocty Ths soluton of the moton equaton of deal 3 Low ampltude waves n sotropc lnearly elastc medum The moton equaton of the lnearly elastc sotropc homogeneous medum s gven by (33) As the f mass forces can be neglected durng the analyss of the wave soluton, the moton equaton can be wrtten n the followng form s s t grad dv s (35) The s vector space, whch gves the dsplacement feld, can always be decomposed nto the sum of a source-free and a swrl-free vector space s s t s l, (36) where dv s t (37) rot s l (38) Usng the dentty rot rot s grad dv s l l s l 43

44 44 Mhály Dobróka for the s l vector space the followng relatonshp can be wrtten grad dv s l s l Based on ths equaton the st sl st sl (39) t t formula can be derved by usng (35) As the order of the partal dervaton s nterchangeable, st dv t st ensues from (35), e the frst term on the left sde of (39) s source-free as well It can be seen smlarly that the second term n the brackets s swrl-free As the sum of a source-free and a swrl-free vector spaces can be zero only f the two vector spaces are zeros separately, from (39) the and the st st (3) t sl sl (3) t equatons can be deduced, where, (3) Thus the moton equaton gves a separated wave equatons each for the source-free s t the swrl-free s l vector spaces The dsplacement vector potental can be ntroduced based on (37) wth the equaton s rot t, (33) whle (38) can be satsfed trvally f the s l vector space s wrtten as the gradent of the scalar dsplacement potental grad (34) s l The dsplacement feld can be wrtten as and

45 s grad rot based on (36), from (39) the equaton Wave propagaton n elastc and rheologcal meda rot grad t t can be derved It s agan a sum of a source- and a swrl-free vector space, therefore the (35) t and the (36) t equatons must be met, where and are gven by (3) The vector and scalar dsplacement potentals satsfy the (35), (36) wave equatons These equatons as well as the equatons (3) and (3) wrtten for the dsplacements show that two types of body waves can arse n the lnearly elastc homogeneous sotropc medum Based on e t kr the monochromatc plane wave soluton of eq (3) can be wrtten as where tkte r s s e, (37) t t k t (38) The s t vector space satsfes the (37) aulary condton, therefore the dv s s e t equaton should fulflled as well, whereof t t e The (37) descrbes a transverse wave s t propagatng wth velocty Snce dv s t, these waves do not result n volume changes The monochromatc plane wave soluton of (3) has the form of where tkle r s s e, (39) l l 45

46 Mhály Dobróka k l (3) The s l vector space satsfes the (38) aulary condton, thus rot s e s l l l Ths crtera s fulflled f the drectons of dsplacement and the propagaton are parallel The (39) descrbes a longtudnal wave propagatng wth velocty It can be seen from eq (3) that the latter one n the two types of waves propagatng n the Hooke medum s faster In comparson of the longtudnal and transverse waves orgnatng from a common source the longtudnal waves arrve frst to the observaton pont 33 Low ampltude waves n vscous flud After lnearzaton and neglectng the mass forces the (38) Naver-Stokes equaton can be wrtten as v grad p' v grad dv v (3) t Let us stpulate the rot v aulary condton and let us form the dvergence of the equaton! Then wth the commutaton of the parcal dervatves dv v p' t dv v (3) Based on the lnearzed contnuty equaton ' dv v, t from the lnearzed barotropc equaton the p' c h ' equaton can be deduced, wth whch dv v p' c t h Substtutng ths equaton nto (3) 46

47 Wave propagaton n elastc and rheologcal meda p' p' p', c t t h where c In search of the monochromatc plane wave soluton of the equaton n h the form of p' p * e t k e r, the followng comple dsperson relaton can be derved from whch, c k h k (33) c h The comple wave number can be wrtten n the form of where k b a, b (34) c a c h h based on (33) (35) 3 3 In case of water the vscosty s Ns m, c 44 m s, kg m these data 3 s, Wth 8 e for the frequency f 5 Hz Ths nequalty s satsfed n sesmc, acoustc and ultrasonc frequences as well, therefore we can apply seres epanson n (34), (35): h b c h 47

48 Mhály Dobróka Wth these a c c h 3 h * tk e r * ae r tbe r p' p e p e e The longtudnal wave propagates wth sound velocty of h b c n vscous flud and t attenuates wth an absorpton coeffcent of a, the attenuaton coeffcent s proportonal c 4 to the square of the frequency The absorpton coeffcent s a 4 / m h n water at a 4 frequency of f Hz, e the penetraton depth of the wave s d / a 5 km The attenuaton s weak: a<<b, or otherwse the wave length s much smaller than the penetraton depth The transverse wave soluton can be obtaned wth the dv v based on the equatons (3) and (33) v v, t whch soluton s searched n the form of and the v v k * e t k e r, aulary condton dspersve equaton can be deduced For the comple wave number the followng equaton can be wrtten k, so the real wave number s b, the absorpton coeffcent s a The phase velocty of the wave v f s frequency dependent, the wave s dspersve So b 48

49 Wave propagaton n elastc and rheologcal meda transverse waves can be created n vscous flud But these attenuate very strong (a=b) The penetraton depth s d a 4 For eample for a wave wth a frequency of f Hz -5 t s d 7 [m], whch s 8 tmes smaller than for a longtudnal wave wth the same frequency Therefore t can be consdered n sesmc applcatons that the transverse waves play no role n water 34 Low ampltude waves n Kelvn-Vogt medum The moton equaton of the Kelvn-Vogt body can be wrtten n the form of s s t grad dv s v grad dv v based on (335) after neglectng the mass forces In search of the wave soluton wth the dv s =, dv v = aulary condton (transverse wave), then the s s v (36) t equaton can be derved, whch monochromatc plane wave soluton has the form of * tkt e r st st e After substtutng t nto (36) a dsperson relaton smlar to (33) can be obtaned k t, where, Solvng the equaton for the comple wave number k t b a formulas smlar to the equatons (34), (35) can be deduced b (37) 49

50 Mhály Dobróka a (38) The wave s phase velocty v f s frequency dependent, so the transverse waves propagat- b ng n the Kelvn-Vogt medum show dsperson and ther absorpton coeffcents s frequency dependent as well In a low frequency border-lne case epanson the equatons (37), (38) can be rewrtten as b 4 a Thus n the frst appromaton the v f b Then wth seres phase velocty s frequency dependent, e there s no dsperson, the absorpton coeffcent depends on the square of the frequency The Kelvn-Vogt medum changes to Hooke body n low frequences n pont of vew of wave propagaton velocty, but t preserves the propertes of the Newton body wth respect to the absorpton At hgh frequency In ths case the equatons (37), (38) lead to the results revewed at the Newton body b, a b The Kelvn-Vogt body gves back the Newton body n the hgh frequency border-lne case Ths can be epected from the structure of the model shown n Fgure The Kelvn-Vogt body s not sutable to descrbe weakly attenuatng waves at hgh frequences Longtudnal wave can be dscussed by specfyng the rot s =, rot v = aulary condtons As n ths case the rot rot s = grad dv s - s equaton s satsfed, from the moton equaton 5

51 Wave propagaton n elastc and rheologcal meda 5 v s t s Wrtng the tme dependence n the form of t e the followng equaton can be formulated s t s, (39) where Based on the equaton the * *, formulas can be ntroduced for the comple Lamé constants Wth these the (39) can be rewrtten as s t s * * In search of the wave soluton n the form of monochromatc plane wave e r k t * l e s s based on (39) the followng comple dspersve equaton can be derved l k For the a b k l wave number results smlar to (37) and (38) can be obtaned b, a, from whch the frequency dependent phase velocty f b v and the frequency dependent penetraton depth d

52 Mhály Dobróka can be deduced These formulas have the form of b, a v f, d n the low frequency border-lne case, e the phase velocty has the value characterstc for the Hooke body, the absorpton coeffcent pcks the value characterstc for the Newton body The attenuaton s weak, as s fulflled trvally from the a<<b condton The dsplacement functon s ae r s s e e tbe r (33) It should be understandable that the Kelvn-Vogt model s not approprate for the descrpton of weakly attenuatng longtudnal waves n hgh frequency border-lne case, because n that case b, a b Note that the Kelvn-Vogt body can be appled wth dfferent parameters for the same rocks durng descrpton of rock mechancal and sesmc phenomena The characterstc tme of creepng process n a rock has an order of magntude of hours-days, a smlar order of magntude of retardaton tme belongs to ths However for descrpton of weak attenuaton 3 5 of sesmc waves the ( )(sec) value s sutable Ths fact suggests that the Kelvn-Vogt model has an appromate valdty, the parameter ncluded n t s frequency dependent and t can be consdered constant only n a narrow frequency range However we can conclude that ths model s sutable for descrpton of weak attenuaton of longtudnal and transverse waves Ths s the reason that s wdely used n sesmcs mostly for the descrpton of wave propagaton n rocks wth hgh water and hydrocarbon content The constant Q model The sesmc eperences show that for most rocks the phase velocty s constant regardless to the frequency, but the absorpton coeffcent ncreases n drect proporton wth the frequency 5

53 Wave propagaton n elastc and rheologcal meda v f c (33) a, (33) c Q where Q s the frequency ndependent (constant) qualty factor of rocks Ths rock model s called the constant Q model In case of Hooke body (33) s satsfed, but there s no absorpton For the Kelvn- Vogt body (33) s satsfed at low frequency border-lne case, the absorpton coeffcent can be determned by a, e compared to (33) Q, e the qualty factor s not constant Smlar results are obtaned (at low frequency) for the Poyntng-Thomson body The Mawell model can descrbe sesmc waves n hgh frequency border-lne case, but then a, e based on (33) Q, the qualty factor s proportonal to the frequency Smlar re- sults are obtaned n hgh frequency border-lne case by the Poyntng-Thomson body It s understandable that the body followng the constant Q model can be characterzed by the comple Lamé constants *, ' * (333) f ts materal equaton s assumed n the form smlar to the Hooke body s formula ( and ' are frequency ndependent) * * The here presented dsspatve rock physcal parameters can be defned by the formulas tg( ), ' tg( ' ), 53

54 Mhály Dobróka where s the loss angle (the angle between the stress and the deformaton n case of pure shear(for eample at transverse waves)) ' has smlar nterpretaton n wave theory applcatons for the longtudnal waves Based on the stress tensor above the moton equaton can be wrtten as * s t s * * grad dv s For transverse waves dv s and so from the moton equaton the s * s t equaton can be derved For the monochromatc waves wrtten n the form of t s s e * kt e r usng equaton (333) the followng dsperson equaton can be deduced, k where For the k b a comple wave number the (334) b a ( ) ( ) (335) (336) equatons can be wrtten It s easly understandable that we can speak about weak attenuaton (a<<b) only f Then wth a smple seres epanson from (335), (336) the followng epressons can be derved b a, from whch t can be seen after comparng to 33 that for transverse waves that the qualty factor s really ndependent from the frequency 54

55 Wave propagaton n elastc and rheologcal meda Q, smlar to the phase velocty v f b Stpulatng the rot s aulary condton for longtudnal waves the t * s s * equaton can be deduced based on (334) In search for ts monochromatc plane wave soluton n the form of t s s e * kl e r the followng dsperson equaton can be obtaned, k ' where Ths equaton changes nto (335) wth the substtuton of,, therefore ts soluton can be wrtten drectly accordng to (335) and (336) b ( ) (337) a ( ) (338) Weak attenuaton happens only f (e, ' ), therefore based on (337), (338) b, a Based on the comparson to (33) the qualty factor for longtudnal waves can be obtaned n the form Q ' The constant Q model ganed wdespread applcaton n the feld of sesmcs and acoustcs 55

56 Mhály Dobróka References Asszony Cs, Rchter R 975: Bevezetés a kőzetmechana reológa elméletébe Nehézpar Mnsztérum Továbbképző Központja, Budapest 56

57 4 Elastc wave propagaton The pressure dependence of elastc propertes of acoustc waves s an etensvely eplored rock physcal problem because pressure strongly nfluences the mechancal and transport propertes of rocks, such as acoustc velocty, porosty, permeablty and resstvty By usng the dscussed elastc medum models n the prevous sectons petrophyscal models can be developed whch descrbe the pressure dependence of elastc propertes of rocks (prmarly the velocty-pressure and qualty factor-pressure relatonshps) By the knowledge of the characterstcs of elastc waves - f ths dependence can be reversed - stress state of rocks can be determned ndrectly based on sesmc/acoustc measurements n laboratory To relate changes n sesmc attrbutes to reservor condtons, a thorough understandng of pressure effects on rock propertes s essental 4 Descrbng the presuure dependence of longtudnal wave velocty The velocty of acoustc waves propagatng n dfferent rocks under varous confnng pressure condtons (Wylle et al 958, Stacey 976, Prasad and Manghnan 997, Kng 9) and also under dfferent pore pressures (Nur and Smmons 969, Yu et al 993, Darot and Reuschlé, He and Schmtt 6) were nvestgated for several decades by many researchers Accordng to general observatons, larger propagaton veloctes are measured on water-saturated samples than on dry or gas-saturated ones (Toksöz et al 976) and the P wave velocty s larger n coarse graned, sedmentary rocks than that of n fne graned samples (Prasad and Messner 99) The phenomenon that the wave velocty ncreases wth pressure s well-known and has been eplaned on varous rock mechancal studes (Wylle et al 956, Brch 96) One of the most frequently used mechansms for eplanng the phenomenon s based on the closure of mcrocracks n rocks under the change of pressure (Holt et al 997; Best 997; Hassan and Vega 9; Sengun et al ) Sngh et al (6) created an emprcal model for the pressure dependent wave velocty after observng measured P and S wave veloctes on several sandstone samples Prasad () studed the same relatonshp for gassaturated and pressurzed zones from the rato of propagaton veloctes of P and S waves Accordng to petrophyscal models and epermental results, we can nfer to the sze of emergng tensons n rocks, and even probably to ts dependence of drecton usng measured longtudnal and transversal wave velocty data Several petrophyscal models can be found n the lterature, eg, Bot model (Bot 956a, Bot 956b), Gassmann model (Gassmann 95), Contact radus model (Duffy and Mndln 957), Frcton model (Wnkler and Nur 57

58 Judt Somogyné Molnár 98, Stewart et al 983), etc These models provde proper approach for the descrpton of the phenomena dependng on the type of rock The Bot model for eample descrbes wave propagaton n a two phase system, a porous elastc frame and a vscous, ncompressble pore flud The propagaton and attenuaton of the wave s ascrbed to relatve moton between the frame and ts pore flud In the lecture notes we present a new approach for the quanttatve descrpton of the pressure dependence of phase velocty of acoustc waves Our consderatons are based on the mechansm that mcrocracks are closng wth ncreasng pressure 58 4 The pressure dependent acoustc velocty model Modelng plays an mportant role n the cognton of natural scence In the eplanaton of a phenomena we consder the most mportant and most essental propertes and neglect all the other (n other aspects may be mportant) characterstcs Hence, we set up a model n whch we smplfy the studed structure and henceforth we talk about the propertes of the model Ths approach was followed at the development of the pressure dependent velocty model The response of rock to stress depends on ts mcrostructure and consttuent mnerals, whch s manfested n pressure dependence of velocty of elastc waves Several qualtatve deas est descrbng the pressure dependence of sesmc velocty Such as that pore volume reduces wth ncreasng pressure, thus ncreasng velocty can be measured (Brch 96) Followng Brace and Walsh (964) we assume that the man factor determnng the stress dependence of the wave propagaton velocty s the closure of the mcrocracks For ths reason, we ntroduce parameter N as the number of open mcrocracks Acceptng ths qualtatve dea a rock physcal model whch s vald only n the reversble (elastc) range - was developed usng the followng formulaton (In our consderatons we focus on unaal stress state and longtudnal acoustc waves) If we create a stress ncrease dσ n the rock, we fnd that dn (the change of the number of open mcrocracks) s drectly proportonal to the appled stress ncrease dσ At the same tme dn s drectly proportonal to N We can unfy both assumptons n the followng dfferental equaton as dn Nd, () where λ s a new materal dependent petrophyscal constant In Eq () the negatve sgn represents that at ncreasng stress - wth closng mcrocracks - the number of the open mcrocracks decreases The soluton of Eq () s

59 N N ep( ), () where N s the number of the open mcrocracks at stress-free state (σ = ) Another assumpton s a lnear relatonshp between the propagaton velocty change dν - due to pressure ncrement dσ - and dn dv dn, (3) where α s another proportonalty factor (materal qualty dependent constant) The negatve sgn represents that the velocty s ncreasng wth decreasng number of cracks Combnng Eq (3) wth Eq () and Eq () we obtan dv N ep( ) d (4) Solvng the upper dfferental equaton we have v K N ep( ) (5) where K s an ntegraton constant At stress-free state (σ = ) the propagaton velocty ν can be measured and computed from Eq (5) as ν =K-αN Hence, we obtan the ntegraton constant: K=ν +αn Wth ths result and the ntroducton of Δv = αn, Eq (5) can be rewrtten n the followng form v v v( ep( )) (6) Eq (6) provdes a theoretcal connecton between the propagaton velocty and rock pressure The model equaton shows that the propagaton velocty - as a functon of stress - starts from v and ncreases up to the v ma=v +Δv value accordng to the functon of -ep(-λσ) Thus, the value Δv = v ma -v specfes a velocty range n whch the propagaton velocty can vary from stress-free state up to the state characterzed by hgh rock pressure The velocty reaches ts lmt v ma at hgh stress values Certanly t s only vald n the framework of the model assumptons (reversble range), because n the range of hgh stresses new mcrocracks can arse n the rock (Ths phenomenon s outsde of our present consderatons In order not to eceed reversble range and to avod creatng new cracks, samples were loaded durng our measurements only up to one thrd of the crtcal unaal strength) Snce λ s a new petrophyscal parameter (materal characterstc) t s necessary to gve ts physcal meanng Introducng the notaton u = v ma -v, wherewth Eq (6) can be wrtten also n the form u v ep( ) (7) 59

60 Judt Somogyné Molnár It can be seen that at the characterstc stress σ* (when λσ*=) the quantty v ma -v decreases from ts Δv ntal value to Δv/e The λ petrophyscal characterstc (materal constant) s the recprocal value of σ* On the other hand, we can also gve another meanng of parameter λ The eperences show that rocks show dfferent velocty response to the same change n the rock pressure or n other words the velocty shows dfferent senstvty to pressure It s nterestng to see, what amount of (relatve) velocty change can be measured as a consequence of a certan (for eample unt) change n the stress For smlar purpose the senstvty functons are etensvely used n the sesmc (Dobróka 987), the geoelectrc (Gyula 989), electromagnetc (Szala and Szarka 8) and well-loggng (Dobróka and Szabó ) lterature So, we ntroduce the (logarthmc) stress senstvty of the u = v ma -v velocty as du d ln( u ) S( ) (8) u d d Usng Eq (7) t can be seen that d ln( u ) S, (9) d whch shows that the λ petrophyscal characterstc s the logarthmc stress senstvty of the u = v ma -v velocty It can be seen that n our petrophyscal model the logarthmc stress senstvty s ndependent of the stress 4 Epermental settng, technque and samples The pulse transmsson technque (Toksöz et al 979) was used for P wave velocty measurements The epermental set-up (Fg ) was compled at the Department of Geophyscs (Unversty of Mskolc) Rock samples subjected to unaal stress were analyzed wth an electromechancal pressng devce and wave veloctes - as a functon of pressure - were measured at adjonng pressures An mportant queston s that how reproducble the measurements are Hence we made tme-lapse measurements n case of several samples One typcal test result s presented n Fg It was shown that the second measurement provded the same result wth very good appromaton Thus the phenomenon s hghly reproducble and the elastc range was not eceeded (new mcrocracks were not formed under pressure) The specmens used n our studes orgnated from ol-drllng wells We performed wave velocty measurements on several dfferent ar-dred sandstone samples Three typcal 6

61 Elastc wave propagaton test results (sample S, S and S3) are presented n ths lecture notes Table contans the descrpton and depths of our studed samples Fgure : Measurement layout for wave velocty measurements Table : Characterstcs of our epermental samples Sample Descrpton Depth [m] S Fne-, medum-graned sandstone 8 S Fne-graned sandstone 35 S3 Tuffy sandstone 8 6

62 Judt Somogyné Molnár Fgure : Result of repeated measurements for studyng reproducblty 43 Case studes The parameters appearng n the model equaton (Eq (6)) can be determned by processng measurement data based on the method of geophyscal nverson (Dobróka et al 99, Dobróka and Szabó 5) In order to prove the valdty and practcal applcablty of the velocty model ntroduced n Secton, we present the results of the appled lnearzed nverson method for each sample (Table ) The nverse problem s overdetermned, so the Least Squares method (Menke 984) can be effectvely used for solvng t Table : Model parameters estmated by lnearzed nverson Sample ν [m/s] Δν [m/s] λ [/MPa] S 57, 87,8,47 S 56,6 753,5,57 S3 38,9 476,6,774 6

63 Elastc wave propagaton Wth the estmated parameters the veloctes can be calculated at any pressure by substtutng them nto Eq (6) The results are shown n Fgs 3-5 The contnuous lne shows the calculated velocty-pressure functon whle astersk symbols represent the measured data Fgure 3: Longtudnal wave velocty-pressure functon on sample S (contnuous lne calculated data produced by nverson, astersks measured data) Fgs 3-5 show that the calculated curves are n good accordance wth the measured data whch proves that the petrophyscal model suggested n Eq (6) apples well n practce It can be seen that n the lower pressure regme, the ncrease n velocty wth ncreasng pressure s very steep and nonlnear Ths s due to the closure of mcrocracks, whch dramatcally affects the elastc propertes of rock and thereby the veloctes In the hgher pressure regme, the ncrease n velocty (wth ncreasng pressure) s moderate as fewer numbers of cracks closed The model was also appled on several sandstone samples (fne-, medum-, coarse-graned, pebbly, tuffy etc) durng the research and smlar results were obtaned 63

64 Judt Somogyné Molnár Fgure 4: Longtudnal wave velocty-pressure functon on sample S (contnuous lne calculated data produced by nverson, astersks measured data) Fgure 5: Longtudnal wave velocty-pressure functon on sample S3 (contnuous lne calculated data produced by nverson, astersks measured data) 64 As t was mentoned the samples were loaded durng our measurements only up to one thrd of the crtcal unaal strength It was found that λ s connected wth unaal strength:

65 Elastc wave propagaton the hgher the unaal strength of a sample, the smaller the estmated λ s It was also observed that the parameter ν s senstve to rock qualty, as t can be seen n Fg 6 Snce sample S was a fne-, medum-graned and S was fne-graned sandstone, the curves belong to these samples are almost the same But the curve of the tuffy sandstone sample (S3) s located n dfferent velocty range, because t starts from a dfferent ntal velocty (ν ) Fndng connecton between the other parameters and rock qualty requres further nvestgatons Fgure 6: Longtudnal wave veloctes as a functon of pressure for the studed samples (curves calculated data produced by nverson, symbols measured data) For the characterzaton of the accuracy of estmatons, we calculated the measure of fttng accordng to the data msft (D[%]) formula D N N k d d ( m ) ( c ) k k ( c ) dk [%], () where d k (m) s the measured velocty at the k-th pressure and d k (c) s the k-th calculated velocty data, whch can be computed accordng to Eq (6) Table 3 contans the value of data msfts for each sample n the last teraton step 65