|
<br />Fg. 3
<br />Vertical stress ~ - .
<br />• = _
<br />a
<br />
<br />I E -39K+G f1)
<br />
<br /> R'= 1.34x7010 (dP~a ~ '- .. .. (3)',
<br />
<br /> R,-K
<br />a
<br />(6)
<br /> tang
<br />IOCIty a. =tan P .. - I6)
<br /> Nom enclature . -
<br />~
<br />~
<br />~
<br />~
<br />lsured , ..
<br />.
<br />,,
<br />~. ~
<br />~ ~
<br />. a = Ploned vertical stress Irom o=0 psi to the Intersection of the ,
<br />wave horizontal axis antl either the Straight-line Mohr-Coulomb '
<br />'ferent '. .strength plot or the G plot, psi
<br />wide ~ E
<br />~ = Younq's modulus, psi ~ ~ -'
<br />
<br />'65e5.r G
<br />•~ K =Shear modulus, psi
<br />=Bulk modulus, psi ~ ~ .
<br />~
<br />.vas to R =Combined dynamic modulus, psi , .
<br />. ~ -
<br />~
<br />~
<br />~
<br /> .Po' ~ =Combined dynamic modulus at o=0, psi : ~
<br />'
<br />, ~
<br />acou5- R, . =Combined dynamic modulus at terminal stress, psi ~~ ~ -'
<br />~ dafa = ~ P
<br />' =_Ar51e o! the straight-line MohrCoulomb strengN plot, psi ::.,.;
<br />'
<br />
<br />Jty of ; dt,,,, compressional acoustic wave
<br />_= The average navel lime of a
<br />l
<br />~ "'
<br />'
<br />~'
<br />~ ~ ~
<br />'-~
<br />~~'
<br /> ~ .. through limestone (47.5 µserJ
<br />t)
<br />~
<br />~
<br />.
<br />.
<br />~
<br />by a .;
<br />~tl:.. ~=The average crave( time al a compressional acoustic wave ~;
<br />'
<br />'
<br /> _ - .r
<br />~. Ihrauggh bdne (189 pseUlt) ._~ ., .. 's~-.
<br />>'
<br />~
<br />•
<br /> .~~'dlr' ='Travel time at a compressional
<br />9cou5tic wdve through lime-;=-
<br />hme- , ~ alone al terminal stress, lueUlt ~,'+::~ . `~,:y-`~. ~.: ~:. l~r~ 'r~ <".~
<br />vhich -':$e :=.POrosiry fraction:,t :...............t;:=.~:i=.;::~;+---'~'
<br />"`
<br />`F'
<br />~
<br />~
<br />r"
<br />•: .. ,
<br />:urate .p ;.
<br />.~ c:: •
<br />`
<br />1 . .
<br />. =. Bulk densi c"-~ .i._; .
<br />,;
<br />-:
<br />tY.glu:::F
<br />Veniwl sVess, Psl •-~5+`~:''~3_;,~;i,:&;~~?,:::
<br />.iciest ? o,"' ~= Terminal (vemcal) stress, Psi ~.';
<br />, _: r^f"!;`;, ~°=-: :`" +'
<br />=
<br />'
<br />rock ~ ..r ~.
<br />:; ,
<br />=Shear stress, PSI-'..'-." ..: ,. ,, :.~,, .,'_'
<br />'-~'.!
<br />=
<br />
<br />I ;TO .
<br />%
<br />~ , =Shear strength of the Mohr-Coulomb sVehgth plot at a
<br />0, psi .;;
<br />i
<br />h4li
<br />t
<br />re
<br />s :~:
<br />~ ;. =
<br />~
<br />~
<br />•°
<br />A
<br />l
<br />f
<br />h
<br />V
<br />R
<br />d
<br />l
<br />:
<br />r-con- ::$.~: g
<br />,
<br />,
<br />,.
<br />;
<br />.
<br />ng
<br />e o
<br />e s
<br />a
<br />ne
<br />vs. o p
<br />o
<br />eg
<br />e
<br />t
<br />: =
<br />:
<br />;ailed
<br />g
<br />i
<br />rriti tl relatively rapid rise in velod- level velocity values are ter-
<br /> ry val ues tt'ith increasing mina) velocities, and the
<br />Rg 2 tr str s value where the level
<br />s ess.
<br />The velocity values be-
<br />come level at high stresses
<br />for the stronger rock. The
<br />es
<br />velocity values begin is the
<br />terminal stress.
<br />Because high stress levels
<br />,. ;7,000 - - att.:.:'
<br />`:+~c;;:
<br />-, 6,OOD _
<br />I :.
<br />- - ~ ~~6.000 -
<br />V ° 4,000
<br />a
<br />rn '3.000 ;~::ti
<br />' 2,000
<br />X1,000 -
<br />~ ,
<br />Fiq. 4
<br /> ~~: ' (: <' ,~~. ~_::: TartP~hrom_ the Mohr-Coulomb ~strenglh Ploq r -~
<br /> 'Oecreosrs wilA incrtuinp rpck slrenplA ~ ~' - ~ ~ ~ -~•^~ ~ - "per
<br />were required to close flaws used for a series of tests in
<br />or fractures in the samples «~hich vertical stress 1. as in-
<br />fested, an approximate safe creased until the rock sam-
<br />level of 6,000 psi was used. pies failed. Each test was
<br />Terminal velodty was not conducted at a different lev-
<br />reached in triable material el of lateral stress.
<br />because the rock grains In Fig. ?, the slope of the
<br />crushed when stress leas in- straight line portion of each
<br />creased.' test plot is defined as the
<br />YVzll log data were used to modulus for that lateral
<br />calculate the terminal stress stress condition.
<br />Eor a friable sand= That anal- Dobrin reported that an-
<br />ysis indicated that a theoreli- other way to calculate mod-
<br />cal stress of 6,300 psi was ulus values is with acoustical
<br />required to reach terminal ~+'ave velocity data. There is
<br />velodty and to apply the a relationship between
<br />time-average equation to cal- acoustical wave velodty and
<br />enlace a valid porosity vat- slope of the straight-line
<br />ue.r ~ portion in a plot of snength
<br /> test data.
<br />Velocity plots
<br />The original velodty plots
<br />may reflect the matrix prop-
<br />erties of the rock samples
<br />and not the Flaws or frac-
<br />tures.'
<br />The following model de-
<br />scription may explain these
<br />results, which are based on
<br />rock behavior in a series of
<br />failure tests. .. -
<br />7riaxia] equipment :was
<br />Table 1
<br />:..~ . ;.-
<br />~ _:;,~...,.... :. :.. ;. .,
<br />4+-`.Mohr-Cori
<br />°%
<br />'Sample.~~~~
<br />'. '~, '~ ': Porosity ~~
<br />~
<br />^ -.
<br />~~Bulk denslry,'~~~:
<br />~: r-c
<br />:
<br />[j
<br />~
<br />,
<br />":~ no. '~.+;;~. :.
<br />... t
<br />=fraction. ":' ~ ,
<br />.
<br />.,
<br />:~ 2 g/cc ~ .... .: :: :.:.::.. •.~.
<br />: an
<br />..1 s.. .~.:..
<br />~ ..:,:~. ::: ;::`0.034.: ~ .. 2.71 .. ~~~~. ~.{:-~ :: 7.6:
<br />~
<br />_.:~2 . - :.:0.139` ~
<br />~.. ~,~..-'
<br />.2.43; `
<br />-'1.1,;
<br />'
<br />:
<br />3 ...
<br />' °.0.054
<br />•
<br />: _
<br />.
<br />~ 2.60 ,..,..
<br />,; ;; 1.5:,;
<br />.: a .. .: :.
<br />::::: o.oa7 -... .- .~. , 0.7 ,
<br />2.66
<br />.:' 6 .' `., `. : :': ,' :.:~.-:,`0.039 ~~ .
<br />~..' 2.73 . ;... i. ;a.: .>.`?::. 1.7_.p
<br />'Cab°laled Irom aCOUSOraI wave velocity Eata al a bw atresa level'. ~?::': .
<br />~.I
<br />3,1972 Dec. 28. 1992.OJ8Gps Journal
<br />9T
<br />i
<br />ind
<br />las,
<br />to
<br />rg.
<br />ew
<br />lne5
<br />Cn-
<br />e
<br />6.
<br />ion
<br />~le
<br />ell
<br />u-
<br />Ta~
<br />~F
<br />am
<br />Jo.
<br />T.
<br />yin
<br />sti-
<br />r.I
<br />1
<br />ed
<br />,lt•
<br />- tr,
<br />Ir-
<br />:: h,
<br />es
<br />
<br />Table 2
<br />
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