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differs only slightly from ~. the apparent angle of internal friction - ,'
.-
.. {sin _
+ sio ."', + (I + lin 4> sill ."') 1>:11 2:11/1
+ sin y' I+sin.sin.o
(7SI
In"erlion from Ih;1 relalion and U1l1ation (6S, Ji "ts Ihc followin,nprcuiOIlJ for 9 alld (2" - f1)
sin /I _ COl. s,n v' j F COl 9 _ COl ,oJF
sin (h - 9) _ COl _IF <XII(lo -
0) _
-an.
""
COl ,o/F
F' _ I _ sin' • sin'."'
(67)
(68)
i J/i2 _ 0 or II .. 0. Th;.
COl 2(t/GX2~
- :w)
(81)
UsinlUlWIlionl (78) I nd (81) the war "rain ' i lts ,,0 .nd b O c:an UpressN;n lerms 01 ", 0 .nd H". Thcre
w
(($1I!It
ill
fI{
---/+,,112' II", d2
Selected Works of G. de Josselin de Jong
24' sin _.,.,. .o/F - sin ...J/GX21 - :W) _ Mo lID _ COl
,0+ N '
(82)
63
'"
cos y' ) the masnitude of N ' .... ith respect to M' Ian ~ COS y' determines how R i. 10C!ucd "ith respect to that trian&ic- The tbree different pos_ sibilities arc indicaled as cases (ii~ (iii) and (iv) in Fig. 2 1. They are a. fOlio ..... . (ii)
When R lies on th. section R, R. then
- M'
tan 4> cos
y' ..
N' "
M' tan 4> COS
y'
(85)
(iii) When R Ii .. to the left of R, then
N' > (iv)
+M ' tan~cosy'
N' < - M ' tan
~ Fil. 21. HoriJ.oalal ,..... of FiJ. 12, IbowiDg R in lho _ (I), (ii), (iii) aIM (iy) 2b' sin ~ co.
' "IF + .in
q.('IGK2~
_ M" Ian
Eliminalinl (aO - h O) between fourth of equations (26) gives
-
,.pons "r
20)
If> cos." -
eq~alion
N°
(83)
(81) and the
. in If> WG)(2¥ - 20) _ (20 - 2«» sin If> - N° (84) The quantities MO and N°, IJ defined by eq..ations (79) and (80~ are the oo-ordina'.. of R in the hori:.emisl plane of FiJ. 12. II may be verilied Ihal MO _ PT and N ° _ TR. fig. 21 . ho ..... thi. horizontal plane as viewed from above. Th
" , .. (M ' Ian ~ cos v' + N° )F/2 si n ~ cos b' .. (M' tan ~ 001'" - N' )F/2 si n 4> cos v'
(89)
211- - W ... O CQM (Iii). In this case (2o/r - 20) cannot he zero, because (S6) introduced inlO (83) would then produce I ncaative b' , which violat.. requirement. (13~ A minimum for )2o/r - 20 ) is obtained by chooslns b' 10
Soil Mechanics and Transport in Porous Media
El..A$TO-PLASTIC VERSION OF OOUBLE SLIDING MODEL
do: 1000Iin do: Jonlo G .
be Jm). So 1M solution is ,.,·F/c04.;> 0 /I. _0
tJ· _
(2'" - 20) "n4«~G)
... (+M ' tan.a1S.· - N · ) 0 (2. _ W) $in ¢Is/G) ... ( _ M ' tan '" cos y. - N· );> 0
",
( I ma~
Mathematical dabralion or the double llidin& free /0111;0& model. A,c!tiCC$ of M fC'-Sa 2t, No. 4, S6I-SSlI. de JOlSdin de Jonlo O. (l977b~ Constituli"" rdl tions 10< lho !low oIa .......1.. u....... bly in lhe lim;\ ".Ie of Itml. f>r«. 9r~ 1111. Coo(. Soil M fClt.. Fitt £"(1"(1, T okyo, pp. 87-9S. ()rQcher, A. (1976). An npeOlMnll. inyesl;<ion of 110" rules for ITInu.ar ItIItcrial, usin&optically sen· tili"" 1115. panicb. GftHfCh~lqw 16. No.4. 591 -
.,,-
U9n~ COIISOIl.dllUJ ~nd,aiMJ Ji,«:1 Ji"'p~ SUD' 1t.1I on ""! IUIl/tti clay•. R~arch "'pOrt T 12---82, M alSlthUKIIJ Inll;Wle of Technology. Randolph, M. f . and Wroth C. P. (198n Application of thl: f,ilure SlIle in undrained simple . hear to 1hI: lhaft capacity of driven piles. Gtow:hniqw 31, No.
Ladd, C. C. oft Edit'" L
4·
(91)
REFE RENCES
BtJri n, D. L (197J~ T10t MMl:imit of JDlllttJuJ h>oIl~ ill llot .lJnplt IN"'" 4P_41111 PhD thesis, Uniyersity of Cunbridse. do: Jonelin de Jonlo O. (l9S8~ T1Ic unddinilenm in kinmllliel for friction matcrials. I'roc. Cot{. £tJrf~ Pta"," Probl. BI"IWdr I, 5S-70. do: Jouclin do: Jnnlo G. (1959). SUllies tJJtJ ki....""'flc. in l/w f]. This gives the impression that the stress point P moves along the cone in Fig. 12 (as viewed from above) in a counterclockwise direction. The following
R. E. Gibson, Golder Associates (UK) Ltd On p. 544 of this Paper the Author states that the vector PQ (in Fig. 12) represents ' ... the strain rate imposed on the material in the undrained simple shear test'. It is, however, not clear to me which of the three cases (ii), (iii) and (iv), is referred to here. Comparing Figs, 12 and 21 it would appear
t(6~x - 6;y)/ Go Fig. 22. Adapted stress space witb limit cone, similar to Fig. 12, but SbOWD from other side
565
66
DISCUSStON question then arises: what stress path will P follow in the remaining cases (ii) and (iv) ? Perhaps the Author will favour us with one of his elegant drawings to elucidate this point. Authors' repl y
The case represented in Fig. 12 on p. 544 is indeed a case iii. In order to demonstrate this explicitly the situation of Fig. 12 is redra"'n here in Fig. 22. In this Fig. the adapted stress space with the limit COile is shown from the other side by rotating the space over 180 degrees to permit a better view of the stress paths. The main rate vecto r, as imposed by the undrained simple shear test, is a vector of length p, parallel to the axis of f1•.IG _ f1 j G and in its d ir«:tion. So it points towards the left, here . It is shown only once as the vector P ,Q for the case that the stress point is located in P, . Let the stress path POP'P l be considered in more detail, fi rst. Point Po represents the initial st ress state, in which according to the st resses. indicated on the co·ordinate axes, a,,(O) a, ,{O) - 0 and a;,(O) < f1~(O). According 10 equations (32) this implies q. > q. and so an active, initial situation denoted as case iii in the paper. is involved. The part PoP, is the stress path during the elastic stage. This part has the diroction of the imposed strain rate voctor of length p. The elastic stage ends when the stress point has reached the limit cone in the point p .. This point corresponds to the point P of Fig. 12 in the paper. The tilted plane R, oP,R." is the plane of the a·, bO shea rings. The imposed strain rale vector P,Q of
Selected Works of G. de Josselin de Jong
p is decomposed into a plastic component P,R,". consisting of only /I" shear, and an elastic component R, oQ, parallel 10 the tangent plane of the cone in P,. The elastic component R. oQ can be deoompooed into a ,J.-c:omponenl R:R o p a rallel to the circular cross seclion of the cone and an scomponent R OQ parallel to the line connecting P to the cone vertex V. The vector R:R* points towards the lefl and Ihal gives the stress poinl P a motion in a counterclockwise direction as viewed from above. This agrees with the corrOCI impression of Professor Gibson. The component R*Q points do"'nward and thaI produces a downward movement, in the form of a helical line on the cone surface, that spirals downwards. AI point P, Ihe langent to this line has the di roction of the vector R, oQ. Two other of those helical lines arc shown on the far side of the cone surface. BOlh belong to case iii initial stress states. Case iv initial stress states produce similar stress paths, that are shown on thi s side of the cone surface. The helical lines on both sides of the cone are similar, but they are each olher mirror image. When the ct:nt~ angle 21{1, of the radius to P , has values between (- tit + i/.) and ( -til - (1,). then case ii initial stress sta tes are in~olved, originally. For such a case ii the cor~ sponding dotted stress paths, after reaching Ihe cone rim, turn downward and follow a straigh! line path towards the cone vertex. The ~urved helical stress paths of the cases iii or iv initial stress states have the border li nes between case ii and cases iii or iv as asymptotes. length
67
Reprinted (rom Geotet:imique, June 1978
TECHNICAL NOTES
Improvement of the lowerbound solution for the vertical cut off in a cohesive, frictionless soil O. DE JOSSELIN DE l ONG· INTRODUCTION
One of the classical problems in soil mechanics is the determination of the depth h, to which a soil can be excavated by a vertical cut off before collapse OC(:urs. When the soil is rigid plastic with cohesion e and no internal friction, - 0, and its flow properties obey normality, the upper and lowerbound theorems of plasticity are applicable to h. Let a parameter « be defined by
h
~
«elY
where y is the proper weight or the soil, and let «coo, be the value of « correspondi ng to the collapse height hooL , . Then according to the upperbound theorem a value of « larger t han «ooL' is obtained by computing h with a kinematically admissible velocity fie ld, and according to the lowerbound theorem a value 0(
73
2
G.DEJONO
conditions for an c)(tremum and to show the kind of disappointments that are encountered when the calculus of variations is applied to determine slip lines. In order to be specific and to deal with ex plicit results, the relatively simple case is treated here
of a vertical cut off in cohesive, frictionless, non-dilalanl soil. The collapse behaviour of the vertical cut offin such a soil has been studied by reliable procedures based on the rigorous proofs of the theory of plasticity. The exact solution is not yet available, but il is known thaI the collapse he ight h'911 is unique and expressed in terms of the cohesion c and the specific weight Y. is between the following limits 3'64c1l' ~ h'Q" < 3·83c11
The upper limit corresponds to a Fel!cnius solution with a circular slip line, satisfying a kinematically admissible velocity field . The lower limit corresponds to a statically admissi ble stress distribution determined numerically by Pastor (1978). Because of this information on the collapse height, the vertical cut off is an appropriate example for testing the determination of slip lines by variational methods. CONCEPTS BASIC TO THE ANA LYSIS
In its simplest form the calculus of variations is used to de termine the shape of one particular line, wh ich is called extremal, because it can produce an extreme value ofa definite integral. The analysis described in this Paper is an application of this simplest form of variational calculus to the slope stability problem of establishing a safe estimate for the height h of a vertical cut off. The one line ana lysis is based on a few concepts that are decisive for the formulation of the problem and are determinative for the resulting solution. The first concept is the assum ption that at collapse, there exists one particular slip line, which is called here the real slip line. The second concept is a class of potential slip lines, defined in such a manner that it includes the real slip line. The third concept is the presumption that a safe estimateofh can be found by determining the extremal. Real slip line
In the plane strain case of an embankment in z direction, the slip line in a verlical X,)I plane represents a failure plane perpendicular to X,)I. Fai lure planes exist at collapse. They are required 10 separate soil masses that slide with respect to each other and with respect to the stationary soil mass below. When sliding occurs over the failure planes, the stresses along it are in the limit state of stress. Since the soil is taken to be frictionless and non-dilalan t, the lines representing the failu re planes in the two-d imensional x, y-plane are stress characteristics. There are two famili es of conjugate stress characteristics. For slip to develop, it is enough that sliding takcs place along one of the two conjugate lines. This line should intersect the boundaries in at least two points in order to separate a soil mass that is free to move. For sliding it is necessary that the slip line has a continuous slope. The assumption is that at colla pse there is at least one continuous smooth slip line which corresponds to a stress characteristic and which intersects the boundaries in two points. This line is called here the real slip line. The shear stress along the real slip line has the maximum available value which is equal to the cohesion c for a frictionless soil. Class oj potential slip lines
For the analysis a class oflines is required that contains the real slip line. Thisclass is obtained by selecting lines in such a manner that their stress distribution satisfies tbe same equilibrium conditions as the real slip line. At incipienteollapse, the separated soil mass that is about 10 slide
74
Soil Mechanics and Transport in Porous Media
CALCULUS OF VARIATIONS TO VERTICAL CUT OFF IN COHESIVE, FR ICTIO:-lLESS SOIL
3
is still in equilibrium. Its total weight is balanced by the stresses along the real slip line. This is called the requirement of total equilibrium in the following sections. This requirement can be formulated in the form of definite integralS along the line. Furthermore, the real slip line is a stress characteristic. Therefore the stresses along it satisfy Kotler's equation, which represents local equilibrium for the limit stress state, in a direction tangenria! to the stress characteristic. The requirement of local eq uilibrium is satisfied by introducing Kotter's equation in the integral expressions. The requiro;:ments oftotal and local equilibrium produce three definite integralsas shown in the section headed' Determination of the integrals'. The value of one of these integrals corresponds to the he ight of the cut off. The other two are zero. All lines that give these values to the inlegrals are called potential slip lines, because each of them might be the real slip linc. Extremal The calculus of va riations provides methods to find the extremal. That is a particular line of the class of potential slip lines, computed in such a manner, that it can give an extreme value to h, the height of the cut off. In the section headed' Determination of the extremal', the solution for this line is produced by standard variational procedures. The extremal of the potential slip lines is not necessarily the real slip line. It can be any line satisfying the equilibrium I;onditions imposed. The value of h, I;omputed for the extremal is called homr here. This value is an extremum for h, if it is a maximum or a minimum. If it is a minimum, this indicates tbat hom, is smaller than the values of h corresponding to all other lines of the class investigated. If this class is large enough to contain the real slip line (the line that is assumed to correspond to h'~lI)' it may be concluded thaI hom. is smaller than or eq ual to h,.rr. The analysis then procures a safe estimate for the building height of the embankment. The analysis presented here was initiated originally in the supposition that a minimum would result. VerificaliQn
The basic assumptions mentioned in this section may occur to be plausible for soil mechanical inves tigators because they are commonly postulated in slip line analysis. There are two improvements with respect to common procedures: the shape of the line is a result of the analysis and the stress distribution along the slip line satisfies limit stress state conditions by applying Kilner's equation. The assumptions are, however, not self-evident and are even debatable. This is revealed by a verification of the solution using procedures that are standard in the calculus of variations. The character of the extremum is established in the section headed' Investigation of the $Clution' by use of these verifkation methods, Showing that either a weak maximum or no extremum at all is involved. This is disappointing because the conclusion can only be that the analysis was not meaningful. Unfortunately, it is not possible to establish, apriori, whether thc basic assumptions underlying the analysis will lead to a meaningful end result. It is only after the solution has been obtained that its character can be verified. Therefore, first thc ana lysis, based on the three plausible assumptions mentioned in this section. is carried out in the following text. In the final section the result is discussed and the basic assumptions are reconsidered. DETERM INATION OF THE INTEGRALS
Theverticalcutoff is shown in Fig. I . The soil is limited by thefollowingstress free boundaries:
Selected Works of G. de Josselin de Jong
75
4
O. DE
Y, ..
"
-'-'~
,.
" ........... C
"
- 1;~:WdV»!)}AV;;«
,
/
A.
SO~O
•o
•
, A .. ,
Fig. I.
Vtrtlcal cut off wiCh p,mntiaJ slip liM
the vertical plane Be and the horizontal planes AS and CD. The problem is to determine an acce pta ble height II of the cuI off. The soil properties involved are the specific weight y. the co hesion c and a vanis hing angle of internal friction cP _ O. No external loads are acting on the boundary A BCD. Failure is due only to the weight ofthc sliding soil mass. Class of potential slip lines
If the class oflincs considered is to consist of potential sli p lines, the li nes have to intersect the free surface in two points, E and F. The lowe r point E will coincide wi th the lower corner point B, and F is on the upper surface CD. So the lines in the ana lysis are lines SF. The shape of the line BF will be treated in parametric form, such that the horizontal x coordinate and the vertical y coordinate a re both functions of a parameter (( that increases from ((a in B to <XF in F. This can be written as (1)
In the analysis the parameter a will be taken to be the local angle between the line and the x direction (see Fig. I). Differentiati on with respect to a is indicated by a prime such that
x' =
dx/d~,
x" _ d 2 x /d'l\
y' =
dy/d~
y. = d 2y /dct?
(2a) (2b)
A small element of BF with lengt h ds subtcnds in horizontal and vertical directions distances dx, dy (see Fig. 2) given by dx=coSads, dy=sin((ds . (3)
In order to be a potential slip line, it is necessary that the shear stress acting on the line BF has the maximum available valu e, which is eq ual to the cohesio n c. The normal stress on BF will be denoted as p and is taken positive for compression. These quantities will be used to evaluate the integrals that are developed in the following.
76
Soil Mechanics and Transport in Porous Media
CALC ULUS Of VAR IATIONS TO VERTI CAL CUT Off IN COHESIVE, fRICfIONLESS SOil
rom~
Fig. 2. For.,., d R (black arrow) aCllng on t"'mml ds 15 JHHM1II p tis ( .. hlle a rro..,.)
5
of a , h... r compO"',,"1 cds and a !>Oronal com-
Total equilibrium
In the state of incipient failure, the soil and aU its subdivisions satisfy equilibriu m. So the forces acting on the part RCF, separated from the main soil body by the line SF in Fig. I, form an equilibrium system. This means that the force Q created by the weight of the part RCF is in equilibrium with the resistive force R due to the stresses acting along SF. This equilibrium wiU be called total equili brium because it refers to the equilibri um of the total mass BCF, in contrast to local equilibrium which is considered later and refers to the equil ibrium at all points ofthe line BF.
Total equili bri um is satisfied when the forces Q and R annihilate each other. EKpressed in their x , ycompOnenlS: X", YQ ; X R • Y" and their moments M Q, M" around the origin of coordinates, this requires that (4)
Since the soil weight acts only in vertical direction downwards, the horizontal component XQ of Q is zero and the vertical component can be obtained by integrating over slices of width dx as shown in Fig. I, th is gives XQ= O,
yQ = - J:}-(Yr - Y)dX
(5)
The moment of Q around the origin, positive for counter--clockwise rotation, is MQ
= -
f:
Y(yF - y)xdx
(6)
In these cJtpressions x, y are th e coordinates of the line SF an d integration is along SF. Thc components of the resistive force R can also be expressed in the form of integrals along BF. Let dX", dYR be the x, y components of the force d R (black arrow in Fig. 2) produced by the stresses on the stri p of unit width corresponding 10 the line clement ds. This elementary force dR consists of a component c ds tangential to ds and a normal component pds (white arrows in Fig. 2). Decomposed into x, y directions these give according to Fig. 2 and using equation (3)
dX" _ c cos a.ds - p sin r:zds = c dx - pdy
(7a)
dY,, _ csina.ds+pcosr:zds _ cdy+pdx
(7b)
From these the components X". Y" of R are obtained by integration along BF giving
X" -
J: s: dX" -
Selected Works of G. de Josselin de Jong
(cdx-pdy)
(8a)
77
6
G. DEJONG
YA =
J: J: J: dYR =
(cdY+pdx)
(Sb)
The momen t Mil orR around the origin is
MR =
J:
(xdY,II- ydX,II) =
[c(xdy - ydx)+p(xdx+ ydy)]
(9)
The three equations (4), that represent total equilibrium, can now be clIpressed in the form of integra ls. T he fi rst two a rc transformed into the relations ( lOa) a nd (lOb) by use ofthc equations (5) and (8). The last becomes relation (JOe) by using equations (6) and (9). This gives
f:
(cdx-pdy) = 0
(lOa)
(cdy+pdx -Y(YF- y)dx] = 0
(lOb)
[c(xdy-ydx) +p(x dx+ ydy)-rtYF- y)xdx] = 0
(JOe)
f:
f:
These relations represent tota l equilibrium and agree with the equation (5) in the paper by Bake r and Ga rber ( 1979). The analysis continues in the following sections by introducing Kotler's equation along tbe line SF. T his is an essential difference betwecn that paper and the analysis followed hcre. Local equilibrium
In a frictionless mate rial (for which tP .. O)a potential slip line coincides with a stress characteristic. Local equilibrium along stress characteristics is e1:pressed by two Kotter's equations, one for each ofthc two conjugate stress characteristics. The equation corresponding to the line BF is (dp /ds) - 2c(drx/ds)+., si n a = 0 .
( II )
Thisisa differential equation that can be integrated along.)". Adj usti ng the integration constant to fit the si tu at ion in point F gives af"ter integrat ion (12)
in which PF' a F an d YF stand for va lues of t he q uanti ties p, a and y respectively in the point F. Let PF be defined by (13) PF = (PF-2C1f.F)/2c then p can be solved from equation (12) to give P = 2cW r +a) + }"(YF - y)
(14)
It may be rema rked here that it is impossible to solve for P in a similar manner from Kotter's equation, when the soil has internal friction, such that tP is unequal to zero. Therefore, the analysis developed below cannot be applied dircclly for soils with internal friction. Eq uation (14) is used to eli minate p from the integral expressions (10). This gi ves then
f:
78
[cdx - 2c(.8r+ a) dy- Y(Yr- y)dyJ = 0
( ISa)
[cdy+2cWF +a)dx] = 0
(ISb)
f:
Soil Mechanics and Transport in Porous Media
CALCULUS OF VAR IATIONS TO VERTICAL CtTT OFF IN COIIESIVE, FRICTIONLESS SOIL
J:
7
[C(X d), - )'dx)+ 2c(jJ .. +a.) (x dx+ )'d)') + ~(yF - )'»'d)'] = 0
(l Sc)
Since (yf - Y.) is equal to h, the height of the vertical cut off, the parts containing 1 become
f: J:
1(Y.. - )')d)' = hh
1
(l6a)
y(YF - )'»'d)' = iYhl(yF +2)'6)
.
(16b)
By taking the origin of coordinates in point G on one-third of the height of the cut off(Fig. I) such that . (17) )'F=ih, )'6= -ih, x. - O the in tegral (16b) vanishes and the integrals (15) reduce to fo :Z
1", =
f1=
J: J: J:
(dx-2(jJ .. +a.)d)'] = 1'1h
1
1c
(18a) (l8b)
[d),+2(jJF+a.)dx]:z 0
.
[(xdy- ),dx)+2(jJf+ a.) (xdx+yd)'» ) - 0
(18c)
At this stage it is convenient to introduce the parametric manner of describing these integrals. From the relations (2a) it follows that dx, dy can be wri tten as functions of the parameter a. in the following manner (19) dx = x'da., dy=y'da. . Further, relations (3) indicate that tan a _ (dy/dx) and using relations (19) this can be written as tan a.". (y'lx') or _ (20) a. = arc tan (y'lx') So the integral expressions (18) can be written as
f
1
=
J:
J: J: J:
0
=
Go(a) da =
f,
=
G,(«)dk)"I> 3. 319. Petrov. Ill. P. (l968~ VarimiOtlol "'~Ihodr in opli",,,,,,
M.",h 26, 1953)
R<SJlOIlO" 01 P"'" ",. .ure ""'er 011 variations in loadin, conditions 01 turnlUndln, "';1 I.....rded by e pain Ikelcton of the IO~ the port! ...ter is incompr5slble, at the instanl of loading no voIu,,", change of the IO~ .iIl take p\a«. This may be ~ by pullinC (U ) where "o is the Poisoon's ratio of the eombined syst ...., water and lOil Ikeleton. As the ",.. tel" iI gnelually u1)tIIed from an element IIlny poi!)t in the loaded toil, the PoiMon's ratio ". al thil poinl dec_ and tendo to the value of " where .«1) is Ihe value of the PoiMon'. ratio of the grain skeleton alone. A result of tome importance concerning tbe value of E· may he derived in the following "'.y. The wato:r mdoled in tbe pores can prt!,-ent volume ch.o.nga. from ottumng, but not &hurin, suains. Thil implies that sha.r s tm. in thc combined syswn is g.rritd by the grain skeleton alone. It follows dim::tly that and hence that the shear moduLi G' and G a rc equal. Then by a "-elI-kn"",,, relation in dalticily theory,
T·_.,
CO. r-.....j 2(1+"O)-E/2(1+.) .. C II all tima,
(4.2)
and in particular
lE" .. P...j(1+.)
at
'00+0.
(4.3)
93
CONSOLIDATION AR OUN D PORE
The relation.. (4.1) and (4.3) enable us to describe the condition at ' " + 0 as R lunction 01 the dasticity constants 01 the grain shlelon E and •. The relations between the stn:sses in the comhined S)'Slem and grain . keleton only, which have already been stated, may be condensed to
-ao_ -a+..
and
T O_
(4.4)
T.
By addmg the nonnal stresses in three perpendicular directions We obtain
- (",' +a,o+",' ).. - (.. ,+",-t ..,)+h>.
PRESSURE METERS
in which a,' etc., indicate th e twincipal stresses in the combmed s)'Stem. With the aid of th.,.., general consideration. we shan proceed to determine the condition.. at 1- +0 in ou r special case.
Under the inlluence 01 the .tr.,.. .ystem (3.2) and the boundary condition lor the ca,-ity (2.2.5), the stress distribution in the wil can be determmed by the aid of stress lunction...' We ""all not enlarge 00 these compu_ tations because they are easy toamyout and by applymg the lo!lowmg e:cpreWon for the radial displaceme"t as derived lrom a .Iress function, ofo:
A. the volume dilatation " which is known 10 be equal to . - (1- 2.)(a,+",+",J/E, (4.5) is zero at 1_ +0, we obtain, by virtue of the fact that
-!.[~-sin~~l l, a, i), , iJr
·;011,
- (","+" .. +",o)- h>- - (a, o+""o+",,,' ) at 1_ +0.
f (")'+
I+.'! [ "
11,_ --1", - --ros¥ Hit', E*
(4.6)
1+.° ,
f,
Z..,•• in\l- · .......d!f.
i. given by
2(7 5.')
(4.9)
effectively the actual water pn:ssu~ ;n the pores 01 the soil. In the case 01 spberically . ymmetriW stress P at mfinity all expressions have a simpler fonn, and it is easy to verify that the boundary condilions (2.2.2) and (3.3) a~ satisfied, together with those of equilibrium and compatibility, by takmg U, _ -[(l+"o)/2E*]{P- IOt)(r.'/i') -[(1- 2. 0 )/E*]Pr _ - (P-w,),o'/4Gi',
-aV_ I[(I+.')/2E']( -IS+ ..·.) -[( 1-2.°)/3£°)5)4..,,'.
II,
[(6-S.0l+5(.b'- 5)OOS't](,';, )'-[3- 9 COS't]('';' )' ) S . (4.8)
In this expression .... is the pressure in the water contained in the ca.vity and caused by the volume change ~V, which is, in tum, caused by the displ.a.cement of tbe wall 01 the cavity. We may obtain av by integnlotion over the ou rfa.ce of the !!phere
-~V_
the reader may verify that
(4.7)
(4.10)
(U I)
However, we ha ve a relatwn (1.1 ) betw..,,, lI V and ..... by which we may diminate av obtaining:
w, - lS/II HbG ).
(" .12)
T his result indicates that the cavernous po~ pressun meter l'egiSle ... a presMl ~ of nearly IS"" long"" b is small in comparison witb I/ G. As in the newest type b is 01 the order 01 I/ IOOG, where G is the s1,ear modulus 01 Ihe . Iiff""t day we have to deal with, the error ;nvolved is less than 1 or 2 perbOWD in (4 .6) LhILt the WILler V""".He in the pores of the soil is one-third of the Sum 01 the principal stresses. The~fore, neglecting the influence of the denominator 01 (4.12), it follows that the water prt:ssure, as measured by the cavernous type meter, is
94
(4. 15)
's. Ti"""'ho l
,..Ole, ~,. . .« lIound I"'''''''''' Sphen ""d lPh¢._ O by COm!$l'onding functions 01 the form (5.7). The function. ~ _ o need no alterations, beca.use ;: _ 0 also satisfies (5.5), neither the rotation_ stress-Functions, We shan limil oursekes, how~\'er, to the S ._..1..-' e~p( - qr).
(6.1)
We shall nOI use a stress functiCIIl now, bul u, as basic variable as all Slresses and boundary condition. can be wriuen as functions of N. by virtue of the symmetry. for insla nce i - ..-'it/a.-(u,"», which giv~ alter integration: il, _ -A[(q' )-'+(qr)->] exp(-q. )-Br'.
(6.2)
In orner to describe the boundary conditions with il, we muSI firstly disf'O$(..... ), and whkh after integration becomes (I/ a)i - u.-C,-Cr' . We mayomillh e lerm Cr', which conlributes a waler· How radiating from a point !lOllTCe al Ihe origin and i. irrelevant here. Therciore, il 10110""5 thai (l/a)i -tOl -C or -Gl _ C+(I/ 01 '""mula (6..9).
the intrgral may be expreued in fUlICtiom in tbe form (,.... Fl(. 5):
R..".,.... 01 """" pr-u'" met ..... unit "
< ";,h .... ,. 'ypieot vatu .. 01 (Jaf4b). lenni 0, where P_value of step loading It ..... .., a - (I+(J-~) 1)l2p..
fJ-(I- (I-4}1)1]l2P..
Selected Works of G. de Josselin de Jong
erfCf- 2"~'f· exp(-),'jA, ' -"/4'Y~
•
(_ (5 .1)).
For the de(,nition of,. _ (6.5) and of bl l ,.... (I. I) and (5.1). Puuinl!._}l, and 1_0 the ezp~ (6.9) reduUl 10 (4. 12). Since ~ is Imall in COmparilOn .. ith Ja the expression (6.9) for the rigKI type bttomes appro~ima t ely
....- Pf l - ezp(,.....1j erfe(,.o.J)IJ (6.10)
97
3a/13
Application of Stress Functions to Consolidation Problems Application des Fonctions d'Ai ry aux Problemes de Consolidation by G.
1)£
--.
1bo numbo, of ....... fW> ~~~~~f:"T~~na"t":':~ti':lr!~~~: _"'O. n.: ~ «>d~""" ...1...,;", '0 ,be t;t>;n lici'y for delUmining OIress-d.i>lri· butions is the appli(".Ouion 01 SIr... l\U\wn (""J."..., ... ", w J=, 1953) 1m"" Ihis con· ceptK'" """" 'u B"'·.··. (1941) w"i< "'I_,iyn f'" 'vn .... U~ •• iun ;n Ihree dimensio ...
,wo
,ha,
Le -"Ibn< do> (ollOllons de 1<nsioa DCCdSlirc .. oulIi>aDI pou, leo problCmca lie consolidation, .. qme,ric uial< ..., lie La pr<miOte .. ra",..,.un'11a OI)Q'I.".,...,.,. Ia _ lla l'OIaboo .. Ia tn>isitm< ~, i Ia o:,.,orc ",kes pan of (ho normal ,,~, and boca.,... of iu viscosily (he disli· palioo of ,uptrftuo'" wale, is slow a nd the wa ler pressure dcct"eues p:adually. Sin'ilarly tbe .isco ''''
98
• .. ; ~(111,)
w" _
(~~Jh)
+ (eu,lez)
...
(~)
_ (1Iu,18z)
w. now introdllOO ,he.,..... funclions Eand D in ,hcf"llow· inK ",,"y: 2Gu, " (, - 2,.K31£I&az) - Z(I -I')(~!JF'~;)
2G~, -
(l -
2,.K~f"/iIz'l + 2(, -
By inst:ni"lllhese e.pressions ""04"" obl.3'n:
f~
2G. _ {I 2Gw" _ 2(1
,,)G ~[~) lhe d;,pla<emcn15 in
, 2,,>t'1"£ •
(5) ~PTC$
(.
_,,~ ,"'D
which indica'. how thest .... fullOlions ..... separaled aoconJinl 10 compression . and rota'i"" ......
The: "pe""or '1" in ,be
c:ose of ""ial ,ymmc'ry
it ti"" problem there is Doe addiliooa.l boundary con· di.ion "'hid> ... f... .be ~ ....... r; $0 the third ...... func:lion is nffO; (8) the boundary is impervious $0 the gradient of "OI.«r pross"", oormal to tbc surf"ce is ",ro.
Rillid Sp/>< •• F.mtot and Z _ awlh
.... combination of •• pras.ions 7 OM 8 liIi"" (I _
(l -
")I ~ "'E- ~ "'DI i:r~: ~~z
_ itw/lr ")I v.' ~ V'E + !,or ~(,.!. ;:, '>",D)I .. 8"'la~
0'
If V''I''D _ O. i, it tp""';on fo,,,,",, from S by .Iimination of II. and Z, and by introduelion
( 0)
E, .. {/, .. CRI-·-'J'I'.
w.""opted thefoliowinSnUtio"" 11. .. (r' + .')1 _ Spherical coordioal'; S.+I _ palynomialioqH.asdefined byTA'" (19Sof, p. 136); '1'. " Lell'nd .. polynomial of fir$t kind , T ..... (l9~, p. JIl7): .... B. C _ .,bi' rarycoM,"nlS. adju .. able ated oq""Ia Pis. Consideru.. • rieid .phe.. wi,h R., Ioadtd a l lime , _ 0 wilb a fo to! r in .he Z-di"""ion, .... have to sa,i;fy tbc following boundary condi,ions a, II. _ 110. "The rigidity of .be .phere gi,u ~, _ 0 ..•. (14)
,i""'.ion
Tad,,,,
The fOf« P i. I Uppantd by the .urn of a ll the stresses from
""',.. and ..,il acting loge,ber on .he.urfaoo. This ~\U
,"'2,,11.' $in 'f' d 'I'[.in 'f' .i" +
(Pis) - ~ J.
.lfl1.lcemtain:
")1 (IS)
A ' ""lb =pecliO
deli of Ii
cos '{'(ii, -
"These '''''' coodilio", ref.. to.he lil'"in . kele.oro and .... sulli· cien, in.he.,..., of an elastic mo:>1>0"" .hat Ibis oondi.ion may be .,ti.fotd if D is a $Olu.ion of 0 ... of \be followina equa.ions:
.wo
(12)
FI"Ofl' tbe .. ~ oq",,'ions II and 12 we obtaio 4 types of ....ess function: E,. E,.D.. D,. "The tWO functioM E, and D, ~.., not diff• ..,nt from ,he funetio., F, ""ti$fyi,.. V' F, .. O.~nd if we take E, _ n,. the ..... i... and .If..... deri,.. d from .hem a.., identical 10 tbose derived from f"" .. is euily verified. So lhe.., are ..... n'jaUy J stress funttiom to sa'i$fy .he boundary oonditions.
Selected Works of G. de Josselin de Jong
(A) .. _ 0
"These
( 8)
~/8R_0
(16)
boundary .oodilions a ... satisfied by ,aktdjr; ,be soIu.;""" n with n .. O.~...u.. E, _ "'W' ~""' . D I .. BR. 1>. _ 0 , _ CR- ' th~
( 17)
CompU,a.ion of the disp1aa:ments and ........ =y be pe,fanned by ,be: in.roduelion of tbe ........ func:tio ... in ,he .."ressions S, 7 aod 10. By introduction of tilde st ...... aod displace""",,,, furlher in ,he condi.ions 14, 15 and 16. il i. easy
99
100
Soil Mechanics and Transport in Porous Media
~~! ~ "
l".,:
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80
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+ i.'i ~• 0,1 • ir.~ II +
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§
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t:"1i
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,"~
!!. I ~[[o.1 _ '" ";:po ii "i' - 1 !iil+~1i~~1;:;;" ,;,~ ~ 20~ IIii' -:; ~ ""''t ~+ t:'" aW ~ _ ;!. ""n"Ii ~S ~ + + "! I ~'F ~ H ~ lijf ~ II-'g- I §~
0'
"
:... if .
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i'
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i
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ils, l lf ..
811.tn:
If ...
11(' - 9,o)/(1 - ,.Il! and
u,.q) "
.
f...~-_J'O'."d .he loOIu,iono fQl Ihc varialion of wi,1I limo Ihow &0 b ..t.............. "...,boerneDt at .he .......... 01' _ l!>PIi",,,;"" (, .. 0) and .... dte,-dfect. 1/1 ,he c~'mno .,... of eomPR!1$ibilll.y c,. ... 0). lbil after_ effect i, eq ....1 '0 i of Ihc ini,ial ............... n. for .... riP
n.
....' ...... 01
lItu"""'",
'mllnle"',
_""i...
""'/ww
7JII' .,W.u /0 "'.~ 'M tiotdat.. G _ I.... IU ItrlpfJ b _ of ,IU IlwJy.
of Ik R. C.
~-
...~ na
and llb Ii... the ........... tkmeM for the -"" of Ihc Ioadod uaU , .. 0""" , ... GO, ~ n:spcn;vdy
2G..," 1'/ .'. and 2G., '" 2(1 - ")1'",,. Hen: apia all ....... ~ ....tkmoa, is obuiDod iD the ooMiott torn:Spood~ 10 Ihc dcf.,.."..,ion 01' the soil under _ . 'Ibo . fter-efl"ea i.apiD ""0 for aD iDc:ompr."..;ble ooil
Selected Works of G. de Josselin de Jong
101
DB JOSS!:W' DB Jom:>, PIlOP. Oil. h G .. 1968.
GlouchiqWf: 18, 195-228.
CONSOUDATION MODELS CONSISTING OF AN ASSE~mLY OF VISCOUS ELEMENTS OR A CAVITY CHAi'< NEL NETWORK PROFESSEUR DR IR
G.
DE JOSSELIN DE JONG-
SYNOPSIS A cavity channel ndwork consisting of ma.ny Un r
Fig. 1 (left). Spring d •• hpot element
... ..•. ,
199
I Fig. 2 (right) .
IF
Retardation hmetlon for Une .... prlng daahpot (40)
special aspect is treated first by demonstrating how a continuous distribution of spring dashpot elements can produce a time dependent settlement. Physical muhanism of spring dashpol element
The spring dashpot analogy has been introduced to simulate the response of the solid phase of a soil to an increase of the effective stresses. The dashpot in parallel witll the spring (Fig. I) retards its compression and can therefore be considered to produce a retarding effect similar to that which the physico-chemical bonds between soil particles exercise on the deformation of the grain skeleton. For mathematical convenience the mechanical properties of spring and dashpot are assumed to be linear. This may be a crude approximation to reality, and better mechanisms could be substituted without difficulty when the real phenomena are better understood; a revised formula for the retardation function would then result, but the considerations developed in this Paper would not be altered substantially. Let r be the compressibility of the spring and 1 the fluidity of the dashpot. A reduction in height, of tlle element requires a force in tIle spring and a force (1{I )(dCfdt) in the dashpot. The total force P acting in the element is then
crr
p ""
f+ !~ r 1 dt
,_0
(2)
which can be solved directly. If the initial condition is for 1= 0, then , ., P,[I -exp (- fd)l (3) where p.=l{r. The bracketed expression, the retardation function, will be called F (,.t) , because it is this function which represents the retarding effect of the spring dashpot element (see Fig. 2). Therefore F(p.t ) = l -exp (-,.t) (4) By taking the values of (,.t) to be zero and infinite in this expression it is seen that
°
for ,.t = 0 } (5) for ,.t = a) • Tn general F(p.l) can be any non·decreasing function which exhibits retarding properties. But whatever the mechanism, the possible retardation function will have to obey equations (5) in order to ensure that the settlement is zero at the start and reaches a terminal value at infinity. In order to include also otller possible secular mechanisms, we will retain the generality of F(p.t ) in the analysis as long as the treatment allows. In general, therefore, the settlement of the element will be F(,.t) = F(p.l) - I
,=
PrF(p.l)
(6)
The variable of the function F is tlle product (,J), which is dimensionless, so I" alWlLYS has the dimension of reciprocal t ime. Actually p. is tlle retardat ion parameter whose reciprocal is a measure of the timescale involved in the retardation process. In equation (4) a small value of I" indicatcs;thaqthe clemen(needs a long time belore its compressibility is completely
106
Soil Mechanics and Transport in Porous Media
200
G. DE JOSSELIN DE JONG
mobilized, b&ause F(,.t) -+ 1 only if exp (-fdJ is reduced to a small number, and this is the case if (I't) has be;
.:It(s) "" u'(s)ag(s)Az (68) This expression describes the transformed relation between settlement and effedive stress and involves the use of a compressibility ag(s) which incorporates the time dependence by the function g(s). Although g(5) is defined here by equation (67) as a function of the frequency function m(p.) , it is not necessary to know m{",) if g(s) is required. The dire to '+io:> along thtl straight line AB, it is more convenient to shift the integration path to ACDEFB. The parts AC and FB give no contribution, but there remains the hairpin around the negative s-axis and the residues of the poles on the undulating dotted lines. There arc no other singularities in the region swept by the shifting of the integration path because it can be shown that the function g(s), as obtained from the time settlement observations, behaves regularly up to the line CDEF. The solution then becomes t(t) = (p,j/2:ri)
f
[g(s)/s4J tanh (q1l) exp (s/jds
CD£F
8 · -paR ~ ...
'1:
1
I {C(s.) exp (s ·t) g{s-) cxp (s- t) } (2m_ I)2 I +s+[g'(s+)/g(s+)] + I +s -[g'{s-) /g(s-))
with
s+g{s+) = l.[+li7t(2m-l)J2fH2 rg(r) _ 1.[ _F. . (2m_I))2/R2 g'(s) '"'" dg(s) /ds
I n the normal solution g(s) _ I , then from the integral over CDEF in equation (78) only the circle around the branch point in s~O survives to give 2mH, bccau!;C the integrand becomes equal on both sides of the negative s-axis, CD and EF. The summation over the residues of the poles finally gives the well·known solu tion
C(t)
(2m-
8 · I [ - -2-li I " )' ... ]} _pm { 1 -,";i,,~,(2m_l)2exp
(79)
The solution (78) can be evaluated numerically after separating the real and imaginary parts of s· and s-; the result is not shown here. An impression of t he structure of t(t) can be obtained conveniently by considering the solutions for large and small t. These solutions follow directly from approximations of equation (75) for small and large s respectively and taking t he inverse of these approximations. This yields: Approxjmaljrmjor large t is obtained by developing t anh (qll) from equation (75) in a series ex pression for small values of qH, which, because of equation (74) corresponds to small values of s. The transformed settlement then becomes t(s) z (pdM)g(s)[(qII)-i(qH)3+ . . . ] (SO) T he first term between brackets gives pdHs-'g(s). Th is has the fonn of equation (68) and represents t he transformed settlement as created by an effective stress of magni tude p, bccau!;C the transfoml of p is PIs. Its inversion gives the settlement time relation, when excess pore pressure has vanished, as explained in the past sub-section. F rom the second term can be deduced when excess pore pressure has diminished enough to permit disregarding its influence, by considering the value of s that makes the second t erm small with respect to the first. Approximation fqr smaU t is obtained by developing tanh (4Jl) from equation (75) in an expression valid for large values of qH. This expression is [1- 2 exp (-2qH)]. So retaining only the first term, whieh is equal to unity, the transformed settlement in t he first approximation becomes (81)
This can beevaluated by takingg(s) from equation (69). Considering the region 1/0 1 «5 < 0:1 Us) z [t. r( l +0I)JP40o"]'12r{3h)l~
The inverse of t his is (82)
Selected Works of G. de Josselin de Jong
123
CONSOLIDATION MODELS
217
According to this solution the settlement for small t follows a power time law with t to the power (! + tal. In the spherical compression test it was observed that the power of t was less than a half. 50 matching of test results requires a negative value of a, i.e. a positive power of sin g(s). The mathematical implications of this requirement cannot be explained here in detail, but from the theory of 5tieltjes transfonns it can be deduced that no physically acceptable frequency function mijt) exists which introduced in equation (67) produces a positive power of s in g(s). This incompatibility of theory and test results indicates that the present model of spring dashpot elements immersed ill pore water is illca]lable of reproducing reality. COllclusion
The basic equation of consolidation in its L1.place transfonn for a spring dashpot assembly is identical to the transfonn of the original Terzaghi equation, except the coefficient of consolidation, which contains the compressibility ages) as a function of s. Solution of boundary value problems with secular effects therefore follows the same lines as for consolidation without secular effect as tong as only geometry is involved. Only the final inversion of the result is more elaborate because of ages). For large values of t the solution reduces to the case when the load is carried entirely by effective stresses, as treated in the third section. For small values of t the solution reduces to a fonn which contradicts test results, and, therefore, indicates that the present model of spring dashpot clements immersed in pore water is not entirely acceptable. In the follo,\ing soction a model will be treated that consists of cavities interconnected by channels. For this mode! the transformed basic equation 01 consolidation is shown to be slightly different, in so far as the permeability also is a function of s. This second function of s opens the possibility to choose frequency functions for the physical entities involved in a manlier such that for smalll the settlement also will conform to the test results. CONSOLIDATI ON OF STOCHASTIC ASSEMBLY 01' CAVITY CHANNEL ELEMENTS
The second secular mechanism proposed by Keverling Buisman (1938) consists of a network of compressible pores interconnected by channels. Tn tht! following "n"lysis the stochastic character of such a system will again be introduced by considering the cavities to possess different compressibilities and the channels' different conductivities. The difficulty now arises that the pore pressure itself becomes a stochastic variable, because every pore has its own pore pressure. This difficulty is overcome by introducing an average pore pressure, which is not a physical quantity that can be detennincd in a test, but a convenient mathematical device that permits the continued use of the classical theory of consolidation. For further treatment of consolidation problems the time dependence of compressibility and conductivity are involved. PhysicallllechanisnI of cavity chanllel elem ellt
I n a system of cavities interconnected by channels, every cavity is connected by a number of.channels to cavities surrounding it (Fig. 9). Consider a sample cavity with a number k. When finally agrcat number, N, of cavities is considered, a summation is effected where k runs from 1 to N. The compressibility of the kth cavity is called Pk; this means that the cavity expels a volume Pk of fluid when there is a unit pressure decrease of fluid pressure Uk in this cavity. From the kth cavit y v channels emanate and these channels connect the kth cavity to surrounding cavities numbered m. I n a summation m wilt run from I to v, v having an order 01 magnitude of 10 in a t hree dimensional channel network,
124
Soil Mechanics and Transport in Porous Media
218
G . DE JOSSELIN DE lONG
Cavity ehaD.Dei network coruout.i.ng 01 compre._ alble porn iJ:>tercow>&cwd ra.ndomly
Fig. 9.
The conductivity of a channel connecting the kth cavity to an mth cavity will be denoted by This means that a volume of fluid Aho will flow per unit time t hrough this channel if the pressure difference of the fluids in these cavities (u. -u ..l is unity. If the fluid pressure in the kth cavity changes by an amount d,jk ",, (8u./iJljdt during dI.,
"Ie...
the volume of fluid expelled from the cavity during that time is dV/< _ -podu. - -plc(ou.til/j dt (83) The volume of fluid t hat can be discharged through the surrounding channels during it is,
dV..
~
.
.-. • .-. !
["/c .. (,,/c -u .. ))dt
If there is no gas in the fluid, continuity requires that dV /c=dV., p/c(auk!Ot) =
L:
(84)
This gives
Ak. (U .. -U k}
(85)
In this expression tikand u., can be considered to represent the excess pore pressure. Let the initial excess pore pressure be the same for the different pores, and equal to p. Then the Laplace transfonn of equation (85) is p_(suk-P) -
•
.-. ~
(86)
"k .. (U .. -Uk)
Solving for Uk, the transformed excess pore pressure, yields uk[s+( l/pk)
f
.. . I
"k..] =P+(l IPk)
i ("_ . 11..)
.. _ I
I n this equation two summations 01 different character appear. notation the following abbreviations are introduced
. "h.
(87)
In order to simplify the
P.k ..
(I IPk) ~
(88)
4, -
[.~." .'.l / [H
("')
T hese two quan tities have a physical meaning. The character of P.k' as defined by equation (88), is understood by considering t be case when the kth cavity has an excess pore pressure p at time 1_ 0, and all the surrounding cavities are maintained at zero excess pore pressure. Then equation (87) reduces t o 11k --
P!(S+P.k)
with the inverse "k - pexp(-p.J)
Selected Works of G. de Josselin de Jong
(90)
125
219
CONSOLIDATION MO DELS
From this result it is seen that ,...~ is a relaxation parameter similar to ,... introduced for the spring dashpot elements. It represents the relaxation occurring in the kth cavity, during the decrease of excess pore pressure by discharge of fluid from that cavity towards its surroundings. The character of 12k defmed by eq uation (89) is a pore pressure experienced in the kth cavity as a weighted average of the pore pressures in the surrounding cavities. The weighting {unc· tion is ),h" the conductivity of the connecting channels. The quan tity 12k as defmed by equation (89) will be called the ambienl excess pon presSlire. Substituting equations (88) and (89) in (87) gives (91) This expression indicates the relation between the transformed excess pore pressure in the kth cavity, al'.., 14k or I'.. it is permissible to extract the average of (d .. -d k ) from the summation over .. and N on the left hand side of equation (106) . In order to bring in th e space cO-Qrdinate we introduce the centre of gravity of the volume oc tho •• t of C. vitiO. COrultitUtlng a .... p .... HI1t.tive elOUlentary volum., .. eubeot i1I ••lected. lor 'Which tho t:!Utanee betweOJl tho cavitin ,. and. m i. I in a dirC when 6' is practically equal to the load p, the situation treated in the t hird section for t he spring dashpot assembly. During this secondary period of consolidation the average ambient excess pore pressure is practically UfO, but nevertheless in a certain number of cavities, connected by badly conducting clmnnels to the rest of the network, the excess pore pressure may still be appreciable. Release of water from these badly connected cavities gives the long term settlement in the secondary period of consolidation. During that period the cavity cllannel model behaves iden tically with the spring dashpot model, and t herefore the frequency distribution of the retardation parameters of the constituent elements is the same for both models. For small values of time (t) the solution reduces to a form which can be made to match test results by a proper choice of t he distribution of conductivities of the channels. This possibility. of arranging the cavity channel model in a manner such tlmt it reproduces observed settlement behaviour of soils. indicates t hat the model contains the required properties. I t is not a verification though that the model is t he correct description of the physical mechanism at work. Other mechanisms may lead to identical effects, in the same way as the cavity channel network produced an identical behaviour to the spring dashpot assembly in the secondary period. The adoption of any mechanism can only be assumed to reflect reality if it has been established that the physics of t he microstruct ure produces t hat particular mechanism, It is sunnised t hat in a consolidating soil the secular mechanism is basically of the same kind as t he spring dashpot and cavity channel systems examined. This may be the case as well in the microstructure consisting of t he assembly of individual particles as on the macroscale, when cracks and channels run through a soil mass. Therefore the concepts developed in this study may serve t he fut ure study of micro- and macrost ructure of soils. ACKNOWLEDGEMENT
The Anthor is grateful for the help given by Professor R. E. Gibson in shaping the t heory and the manuscript, by Ir H. W. Hoogstraten in considering the properties of certain integral t ransforms and by I r A. Verruijt in overcoming difficulties at all stages of t his work. The basic ideas of this study were presented in a lecture in February 1966 at King's College, London. REFERENCES L. (1965). Consolidation 01 clay with non·line.lr vi$l;osity . GJ()I~chniq ... IS, N". 4, 345---362. DE l°S.EL',.. DE lm,o;;, G. (1942), J(aflaa)stel$1;lhypothcse . Report. Lab, Grond. Mech. Delft, K 9153 . liE 10'SEL'N 11& 10>is.
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by "",taloalloy>i'~' A
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",. eq.... ,i.ing ofl"oct '" A on , if tho .. ,.,.. putly Tau In Ib" ... $f>OCI. "",,,,,d , hat a bi,ber ,II< ptaIr: ...... al1"""lh tbe dilli0. thaI theoe dfeeu betin to ohow ""ly bo)'(lltd dilution no,ioo of 30'7. Witte"''' ho"""p"'. tin.nU b< d< , .., ",pr.n...ofilluinkinc... n... • i"8 . or !iq... fo.ction o(.ht riol. DiMion. h."..." .. ,. mu" "'" P"'''''o' tho ma,. m1 Invntip ,. d from givUll ....,.,gh .b ....
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va" d« M...... "'''''''I."~ Ex.,... ... "" St .. Ioo. C"",; • ..,.. ); .. h... ta.d .. "!be io indeb'''' ,. H . I..ob< ID~oI
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.'
[(,, - ., -T lo ';;)/('+A"'"' )]
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143
l a/13
A Capacitive Cell Apparatus L'Appareil a Cellule Capacitive by G. DE JosstUl' DIO JO)ti ... if the ,tnoins may be determinod $nIJo tonoeplion of sIIca ..... ~ will be ~""n,
o.t .... l...d".. In Fi,_ I ,he apparatus is schematically represoonled. Tho cylindriaded """"",lIy by ,...ishu and boti· zontoOy by a ir pressure on the rubber •• lhe ""nical ....,.,.""'01 is .... d on ,he d ial pugUn. ll ~ ~~~!:~· j l 01 ii'fit l&::!.a E=·{·!i !i ~~·" 5. ~.
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146
Soil Mechanics and Transport in Porous Media
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All. .unocti"" .... y "r ",prt:$ 1e,.. 1. Startin,g from IhiI poinl A ditr"",m palhs may be foUOWI:'tI Ocpending on tho ed abo ..... Tho path AD of SU>epcst dc$(:enl is foUowed if a doona.I of 7 i:I oo:ompanicd by "" i1l£mlSO f y. This happe", in an _ I " " web as tho DIItch..."U whe", tho horizontal dhpl;l(lO......1 "f tho sample is ....!ricIN by an increase of horiIonlal _ . WIled tho .... nical suess is k.pt COII51anl this oisnif><s
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H""'~.. r in ooot ;, of gnoa. help in the interp..,.. ... tioo f admissible shear "",istaooo.
...ccuted
Study of Sboat
R_
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I • J$Ochromu pOu, 1. c .. d' une fOle. conce,mle alluont sur un domH,jan. compa,h. an. l oochromu !>OUr Ie ca. de d.... dl aqo u eo
(J
de Ja liane Be
,
.. Rt
(0)
sin_ + sln(_+28)
R est Ie rayon du disque, l'an,le de la direction OB avec la direction AB,
oil:
Les Miles _ et 0 sont
consld~r's
WS1Uf, s'lls sont orlentes comme IndlQu;\ Sur la fl,ure,
En pasanl .. " 0 dsns II formule (!», on obUenll a relaUon oblenue p.., FROCHT , ", - 0, vslable pcur un couple de forces dlam~trales. Au point D, sllu,; symetrlQuement par r,ppOrt
a
II et B, oil
P
"n "tlll ....nl obllent :
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~ene .ale (1 ),
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n etant malntenanl un nombre PC
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20
illS
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on. :
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n"cessalrement entier. on
sin 2 .. I + s in ..
In
Celie formule montre Que, d.ns Ie cas lire sent, I'o.dre IsochromatlQue ne s ' annule pas au bord. Nesnmoins, cels n'emll~che P,as I'an abae, parce Que la valeur de no eSI. en prallque. llmlte e pou. les raisons ulva ntes. L'In,le .. est ecalement I'ancle de la force P avec la n",male au plsn de conlsct. Ii ne peul donc depasse! I'anile de frDltement , qui es! d'envl!on 200 pour Ie m~erlau utlll ..e dans lea essala. Lor a que I'anile .. varie entre _ 200 el + 200 , la quanUte s in 2 .. /1 I + sin .. ) Ville enl/e -0,98 el + 0,48 . D.n s In uuls lea forces P sont au maximum de I' ordre de 300 N et ... Issent su r des dlsques de 1,5 em d ~ r&yon. PoP' Ie m~e rlo.u uUlis~ {CR 39), Is conStanl e C eat d' envlron l )( 10-' cm'/ N . L I lonl\1e u. d' oDde ), Him d' envlron 600)( 10 -, cm, II QuanUte PC /( ~ R), ) est toujours Infe . len re i 0.3. LI formule (7) conduit donc i la conclus ion Que la valeu! absolue de n". c ' est-i-dl re de I' ordre 110chromallque en un pclnt du !>oro du dleque 'quldlstanl des pclnle d'apllllcallon deB forces, est toujon" Inferiep!e a 0.3 . Alnsl. pulSque Ie nomb re entier ie plus proche de la . aleur de n D eat toujoYre z~ro, II eSI i. prholr Que Is recherche de I'ia ochrome d'ord re :rero ne prhe nf,era p... bnuf)Oup de d lfflculte dane Ie cas d' un dlsQPe ch..", p.., deux forces dlrilees selon une cords arbltralre. On nole ... Que ce type de sollicita tion est Ie cn Ie plus ,enerll de sYsteme o!qulUbr ~ comJ)Oa~ de deux forces ee ulem enl . En ext •• pclant ce .esuhal, on peul admetl re qu'U eal probable Que 188 conl ralnles i. I. J)l!rlpMrle d' un dls que chllie pa! plualeu!s forcn concentren (cf. Is fl ,u rs 3, BU verao, qui present. un cert ain nombre de dl aQ un charies de c ette ra.;on ) aeronl ".Iement Irh pelitel , e! que I' tsochrome d 'oro,e ~~ro eers siluee i. proxlmil e du bord du dlsque.
- 17 _
152
Soil Mechanics and Transport in Porous Media
J . [)flail de la pho\OIl..,hl . d'ur> elPpll.,..nt ch .. ,~. montrontlu iooch""oe. l l'! n!~r\ .... r du dl${j~e • .
F;,ur~
Qualld 1'llIOChrome d'om re u ro e8e re per
m.
ANALYSE OES FORCES SUR L ES DlSQUES
Comme oous l' avon8 YU. la dIrection d'une torce de conlact est delermlmie par Ie \leu des centres des cercles. et sa valeur absoiue eal d~lermloh Pat Ie dlamelre et I' om re d' une com be lsochromatlque. A lor s, loulea les forces \ransm\aes It l' \nte rleur de I' aaaemblage sont COnnueS en direction IUIui bien qu'en vsleur Bbsolue . II eal alnal poSlilble de v~ r lfler que lea rhuUMa satiatonl Bul cond!t!on a d ' eQulllbre des dl s ques Indlvlduel s . Le pr ocid~ de veTlflcation va ~\re deer!] Itl' alde des flgurea 3. 4 el 5. Lea ftgures 3 el 4 montrenl Ie detail d'un cerlaln empUement char ,~. et I. fl,"", 5 repr~8ente Ie dlBllramme des to rce a 1 co"esl>Ondanl.
c fI,u,," 4 • 1".. Quane dlaque$ au cenl", de la n",re 3 ..ec I.... ra lo.c... de OO
panne.u~
de P e,spex,
Entre les pBnneaux Se I.ouve plac~ I' assemblage de dlsques clrculal.es, fab.IQues i partl. du mate .lau photo .t\lutlQu e CR 39. La repartltlon delsmh de cn dlsques etall 1a sulvante: 6 avalent un d la~ tre de 4,OC"' : 1de3,5cm : 16de3,Ocm ; 33de2,5cm: 33de2,Ocm ; 36de 1,8 cm; 28del,5cm at 31 d e 1,0 em, Ces dlSQucs sonl dlsDQses au hasard daus\a cuvette, ll ' exception de ehacun des quat!e coin s de l'usemblage carre, oil I'on a mls, pour des . aisons p,atlqucs, on grand dlsQue. Pour hite . l ' apparl~Jon, au cou.s de la fab.lcalion, de cont •• lntes Inlllales aux bords des dlsQues, II a fallu opolre, avec beaucoop de soln. L' assemblage peut litre chars' horizontalement et veniealement par des plaques rlgldes de m ~ me ep.lsaeu. que les disQues. La plaque Inferl e u!e eSI fixe.. rlilldemenl ; lea trOIs . utrea plaques sont lIb. es de
- 79 -
154
Soil Mechanics and Transport in Porous Media
fir'" 6 - Pbotolrap.ble d' u" empU ...... ,
dI.I&~.
Le det ..U d. 1. III11.e 3 •• Itw ••
R>
Le fait Que les fo.ces Mte. mlnees .. panlr des phQloiraphlea de I' auemblale oMlssent aux conditions d' eQulllb re des dlsQues lndlvlduels , IndlQue Que la methode utllla~e pour determin er lea fOrCes de contact
f'i,,,,, 8 _ u..r. d~. ro. ou .",rres"".don, . I' .... pll.m..,tdel . Pli. 5. L. lom e A co r· •• ,pond i ceUe de I. PlI, ~
Fi~"
9 _ "por. de. rorc"" cor~.pondant i.l ' ..,plle ",en' dela Pli. 7 .
- 82-
Selected Works of G. de Josselin de Jong
157
esl correcle. De ",e"'e. on consl81e Que Ie rBj)lKlr! de la hauleur 10tBle d' une epure it. sa largeu r correslKlnd bien au rllllport de la fo rce ~t8le verUc8le 1 Is force horl~ootale . cependant les di",...,slons KbSolues des epu res dCpassent en gcncr8lles "a leurs des charges exterl emes d' envlron 10 'ii. Cecl dolt eIre Ie resoltat d' une crr.ur systematique, par exemple d' une dlrterence entre lea proprietea mecanlQues des disQues de I 'empilemcnt ct celles du dlsQue de compa raison . Celt" erreur a ~Ie compensee par une modification des eche!les des epu res.
f',,,,,,. d. ~
10 • DIIpUornenl de la !"i l. 6 ..ce les fore .. oontact. \. '''p&ls.rer d •• ll",~. e OI propOrt[nnelle la ,'&1 .. , abSOlu e du forces.Lalew ....
VU.
_
C O N CL.U S IO N
II a ete montre Que. par I'emplol des methodes de la phOI-elasticHe. it etait pOSS ible de dete rmin e" avec suffiumment de preclalon. les forces de contact dana un emplleme"t de dlsques. Les photographies permette"1 de reconslrulre tes Mplacements relalils des disQues. Qui constitue nl lu deformations leo meIri~ues de l'empilemenl. La mesure de ees dHormations a deja ete faUe par beaocoup de ehercheurs, ii I"aide de l'etude d' un empllement de rouleaux en aluminium. Cependant. des experiences aur un empllement de rouleau~ ne donnent aucune Informllion Su r les fo.ces a l' jnte rleur de ]' empllement. Nous avOnS j' impre&slon Que des recherches. teiles Que celles decriles leI. sont necessaires pour comprendre Ie comportement mecanlQue d ' un emplleme nt de dlsQues.
_ 85-
160
Soil Mechanics and Transport in Porous Media
IJIBLiOGRAl' lIl E
J_ BlAREZ _
Contrlbullon ll'.tude des proprletb mecanlQues des sols et des mate,laux Pulverulents. Thhe de Ooclo,al. Grenoble, 240 pp .. Louis-Jean. Gap. ]962. J. BlAREZ et K. W]ENDIECK. - RemarQue sur I',!,!asttcite etl ' lIllsotrople des materlaux pulverutents. C.R. Acad. 5
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luperimposed com'ection in mean flow direction. Following the classical work of Chandrasckhar he then developed a probability density function describing the probability t hat a particle, after N steps, arrives in a given infinitesimal volume during a given infinitesimal time interval. Once this probability densi ty funct ion is know n, he computed t he longitudinal (flow direction) and transverse standard deviations and ~be[Cby quantifying the longitudinal and trans verse dispcrsivitics in terms of the system parameters.
163
4.1 Introduction to Flow and Transport in Porous Media
,. '00
•
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Figure4.1; Di.!per&ion in a poro1l.! medium. The tmcer UJlI!rime .. t was carned out by Michael Rohr of the Institute for Environmental Physics of the Uni1HlTsity of Heidelberg (2001). We acknowledge Profe88or K. Roth for aI/ouling IU to use thi$ material.
From his approach it is d ear that he considers tracer transport in porous media as a discrete process. Unlike many of his colleagues ill t he field , Jos was never very much in favor of a partial differential equation of convection· diffusion type for the tracers. He describe; transport in terms of integrals based on probability density fUllctious describing t he local phenomena. Much later, in his Socorro (1970) report, he considered uacer dispersion again. In this unpublished work he introduced the concept of Elementary Conveyer Unit (ECU) as opposed to the well-known and much used Representative Elementary Volume (REV). An BCU cal"ries the direction of flow and is char· acterized by repetition and by independence. The latter means that a tracer or fluid partidc has a
/n&titl"in!! movement of both fluids. A ]lllr:"lllel plMe model has been
Selected Works of G. de Josselin de Jong
3741
employed for this purpose by Sallting [ 1951], and sand models have boon used by J/arder, Simp8on, Lau, J/ote3, alld McGauhev, [1953]
and K eukgan [1954]. None of the publieatiollS indicated above C(ln· t:lins a mathematie,"l method by which the cor· rect computation of the movement of both fluida lUI derived from given boundary conditiona e:m be made. The first work which BUg· gested a solution to this problem was an un· published study by Edelman in 1957, 'O rondo wateratroming van een niet homogene vloeistof.' In this study he introduced a concept which proved inspiring for the de\'elopmcnt of the present theory: the replacenlent of the two fluids with their own characte ristics by onc hypothetical fluid wit h the same properties o\'er the entire fidd. In this hypothetical filii!! a. row ef sources coinciding with the posi tion of the interface takes care of the change in properties of the two fluids. EJclman determi ned the source disLribution in such n. lI"ay that the .'elocity field created by these sources is equal \.0 the real velocity dist ribution in one of the two fluids. The velocities in the other fluid can be obtained from the velocities created by the sources by Ihe addition of a fictitious vel ocity. This can· cept was developed only fo r a horilOntal inter· £:\(:c, Il.lld the absencc of sources along a verti· cal intcrf~ce was inferred withollt further proof. The present work [de Jon e/in de Jong, 1959] was based on Ihis coneept of replacing the two different flllids by one hypothetical fluid and of introducing the different fluid properties by singu larities along the interface. In determining the character of the singularities, however, the aim Wall a singularity distribution that would directly create the actu:u velocity distribution in both fluids. By this approach the treatment of the two fluids becomcs equivalent, and !Jc. cause of this equi,·alency the extension to more fluids can be made without further complica. tions. T hc present theory shows tlut it is pas-sible 10 meet these requirements with two kinds of singlila rities, vortices or sou rces and sink6. The choice of the kind to lIJle depends on the objective of the stndy. If the objective is the detennination of the discharges, it is C(lnvenient to introduee vortices. 1f the pressure in the fluids i~ to be dctennined, it is more convenient 10 usc the sources n.nd sink6. Furthennore, in the pres· ent study any inclination of the inlerfnte and
169
"42
C.
lhe cue of a gradual transition wne
D!:
JOSSElJN DIll JONG
~re
eon-
aidercd, 118 well ft!I the introduction of boundary conditionll. The importAnt advantage of the ~ingularities
is the poaibility of rolving any boundnry value problem involving two or more fluids by application of the established methods of potential theo ry for the 1!01ution of boundary value problems of one fluid. Although the singularity method i! completely gene ra! wi th regard to differences of fluid prope rties, the present wo rk derus mostly with the inftuence of differences in density only. The effect of differences in viscosity ctln 11180 be reprcsentOO in terms of aingulll.ritiCII, /Ill indicated, but the procedure is not amenable to mathematical treatment. How an electric lUlalogy can be used to account fo r viscosity dilTe rences in connection with singula rities for the density diD'erenee3 will be shown at tbe
end of Part 2. Ph I/fico! (l.$wmptWII3 oM bo.Iic eqt may be combincd with an auxiliary function 4>, so th~t
_oi> _ _ q. ( 14)
"
Soil Mechanics and Transport in Porous Media
MULTIPLE-FLUID FLOW TH ROUG H POROUS MEDIA The condition of continuity (3) then gives
'V''''
co
(lfi)
0
and from the condition of equilibrium, which fin:!.lly resulted in (12), it follows that
a' ",/ a3 an - ,i/ iJn iJ • .,.
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I' "" '1', I'~rt
-(klp.)(irr/ih)
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(24)
(25)
with the solution (13). F rom th is solution the values of '1', :lnd (iNrl/iJn) elln be com puted along the bou ndaries. In geneml, thClie values ",·ill not coincide with the boundary vslues 'I' or (a'll/iJn) required in the problem. The differenCe!! ('I' - '1',) and !(iNr/an) - (iNr,/an)] then constitute new boundary conditiollll which can be IItltislied by part 1I of the !IOlution. The quantity 'I'll must satisfy the deficiencies of part I at the boundarie!! and (26) over the entire field. The detennination of ifl/ constitutes So boundary value problem, with a unique solution to be obtainod by known methods of potential theory. The addition of (25) and (26) ahows that the superpoeition ('II, 'lI'f) eatisliea the requirement (12). It will prove convement to U!lC image vortices of opposite sign in order to create vaiues of if, = colllitant along impermesble boundaries so that, for part II of the sol ution also, these boundaries remain impermesble. Th e thecr jWw at a... abrupt intliTlace. A Rheet. with ~ine")llrities is known to eont!\in rli~_ eDlltinuitiCll. llllltead of showing this by evaluating the principal Cauchy value of the integral expression in (23) as the interface is approached from either side, a sho rter presentation may be given, using equation 12 as a st~rting IlOint. Gauss's gradien t theorem applied to it for a smn.ll reetangle d~dn straddling the interface gh·es, for .:l ,·anishiogly smail width of dn,
+
JJ "9'\rt
d3 dn .,.
f
(iN< jiln,) ds,
+ f (iN' janJ ds.
(27)
where 11.. and n. are the normals to the faces d~, and d,. pointing toward the exterior of the rectangle. These coordinates are related to n
Soil Mechanics and Transport in Porous Media
MULTI PLE-FLUID FLOW THROUCH POROUS MEDiA
with a direction (Fig. 3) according to
+n
~
+11,
+n
=
9,
-n.
n
'"'
\.
'l.;,\
....
From the surface integral of (12) aud the results of the abrupt interface analysis it follows that
.. '
f [(: ),- (: )J" ~ -; ff ~"dO f "" - ; ("Y. - "y ,)
3747
'"
lI in a d.s
... \. \.
A colllltant along aU the boundaries. From potential theory it ill knol"n that i'" must be consum t if it is to aati.afy these oonditions. If the arbitrary value zero ill 8.!lI!igned to the constant, the solution
Selected Works of G. de Josselin de Jong
== o. In_
stead of computing fl from the vortices in the z == x + ill pilUle of lo"igu re 6, it is more convenient fi rst to introduce a transformation in order to obtain a simpler arnngement of the images than to work with the infinite series necessitated in the z plane. By the transfonna_ tion fu nction, ~ "" (2c!-r) In t (3 7) the infinite strip of Figure 6, is mapped into the half plllne co r responding to pos itive real values of , == + iT} ( Fig. 7). The boundaries are mapped into one vertica! line, the 'I axis Ali'. The interface is transformed into a half ci rcle 8ME with radius I. Fluid I occupie!; the region of the half plane inside the circle and flu id 2 the rest of the half plane outside the circle. The image!; arc located on the [eft half circle BNE and have the opposite sign, in order to make Ali' an impcnneable boundary. T he subscript foero will indiClLte thst the interface is concerned. A [inc S (41)
This integral h8.'l no solution in closed form, and, sinoo the aim of the computation is the movement of the fluid, the specific discharge might aa well be introduced at this moment because it facilit8!.el! the evaluation of the integral. The specific discharge, according to (19), is obtained by a differentiation, and, since it is pennisaible here to carry this out under the integra! sign, (41) yields
178
JONG
Fig. 7. Transfonned infinite confined aquifer in r plane.
+·Cr. - Y.) 2...p
+J.
[J.
u ..
(I
t
'r.10)10
r'r.]
.". (I 10)10 Now integ ration with respect to !IOiution
+ kh.2- I,) IIIn 10 " + [In 10 - In (I -
(2) 4 giVe!! the
t.
In (I - IJ]U1r
10>]".".1 (43) The rea! part, the 2: C(lmponent of the spe.). The vertical component in fluid 2 will therefore be g•• -
-
k(y, - 1') (2'1' • •
t,)
(46)
T
On the interlace the a.ngles "", and of. both have the value 3".j2, so the difference of the parallel diseharge at either Bide of the ioterfaee iB
(g •• - g• .>u •••• ,...) - (l:/~)("Y. -
"t'J
(47)
which is the value eJlpected for the ahear flow a vertical interface (II = %1l') according to
AI;
(28). Tbe e:rpresslona (45) and (46) show that on
the lineI! of equal value for q. the angles "", or "'. are eorurtants; these lines, therefore are ares of circles tbrougb tbe points Band E (Fig. 8). Tra1l!formed to the z plane these Jines for equal q. and q. fonn the pattern of Figure 9. For each point of the aquifer the specific discharge vector can be determined from the components q. and q., as is shown in Figure 10, where arrows represent the flow in magnitude and direction. The advance of the interface iU./dt at the 0 is equal to the eomponcnt of the moment t mean velocity perpendicular to the interface; i.e" in !.his case, the z component of the specifie discharge divided by the porosity. F rom (t4) it foUows, therefore, that
=
Selected Works of G. de Josselin de Jong
k('Y. - "y,)
In
er>,
3751
IEPr'l BP • e
.. k(-y. - "y,) In or"
Itot.-1 + ~I
By use of r. exp (n./'k) '" exp (iry./2c) this reduooe to
d~o
'"
k(-y.~ "y,)
In {cot
[r(c ~
Vo) ]} (48)
Figu re 11 shows thia ,'e1odty as a function of Y., (l.n S-shaped cur.'e which is representative of th e shape of the interf!lee a small time after t = O. Di~placement of Ihe interface nt t > O. As the interface turns, the shape continues to be in the fonn of an S which is tangent to the imperme(l.ble boundaries. For all subsequent positions a different flow pattern applies, because the vortices, which generate the flolV pattern, shift position with the interface. The detennination of the advancement of the interface necessitates the computation of the specific discharge component perpendicular to the interface. Also this eomponent changes with the shifting interface. Since an exact mathematical !lOlution of this diBplaeement was not obtained, a first approximation is given here, treating the rotation in the center point of a st raight-lined interface. Let tbe inclination of the interface with the hori~ontal be .. a rc cot a, !lO that the interface equation is
=
Zo .. VO cot II - avo
(49)
Then a line segment dz. of the inte rface is equal to
cUo ... dz.
+ i dyo
"" (a
+ i) dyo
(60)
A comparison with (39) ~how8 the only dif· ference to be the replacement of i by a factor (a +i). T he vortex stre ngth associated with a line segment dC. becomes, by aimilar reason ing,
+(2ck(-y. - "Y,)/(a
+ i}.."ro] dr.
(5 1)
The image interface is given by the equation (m = integer) Zo .,. 1I[(4 m
+ 2)c -
yJ
179
G.
3752
Fig.S.
Line~
JOSSELIN
DE
N:
JONG
of coll8tant q. and q. in r pinneo Fig. 9. Lin"" of con~tant q. Rnd q. in z pl=e. Vlllues indicate 12 Q. P/k(~,) and 12 Q. p/k(-Yrr,) , respectively.
. _...."_..._-
. ... ... / 1 ,,, ........ ..... ..
,, - o - - ' - ' - ~ - ~ - ~ - - . - --::r . --
"' - $ .'
~/
, FlUIO!
,
0
, , ,t /
\ t \
, ,\
, "
0 _
0_
, ,_ + _ .
'10. "" "
'
..
•• •
I t \ I I I
I , ,
, ,
........ ".;:: _ _ ' _' _ _ " _ 0 _
,, ,
I " ~ " ",
M"' _
., "
_ _ ~ ..... _ ... _ ... _
4
=
l'ig. 10. Di.,ehtlrge vectors at lime I >
a > 80~ for Figure 13 and 77· > tf > 68 0 for Figure 14. In the p ictures the area swcpt by the interface can be observed as a lighUl r band, covered by particle tracings of cTOSl!ing directions. These cross paths nrc created by the shcar 1101" at the intcrfaec and are :1.lso present in the rulalytically ob~ined 1I0w pattern of Figure 10. In both figures thc S-shllped interfllce predicted by Figure II i! dist.inguishIIble, and the inelin.'ltion at the center follows the computed ,·alues approximately. Electric analollY. I n the electric model!, which a rc rmtlcu!.'lrly .1pproprinte for the plotting of potential lines and strcfUIllines [Malavard, 1!)5(;], it is impossible to simulate diffcrences of specific weight directly. Howover,
181
G.
3754
1111
JOSSE LIN
01
JONG
"
•
+lJ
~'
~
EE
t .. \ . ·······H ,
"
:, ,. ,
..' ~ .' ....
,
....
-;
...
\~.:.-
'\
'I-,~-.-.
!
\
Z· prAM
'"
1\
/",
( _r'4KC
"
Filii:. 12. Slope of rectilinear illterfac:e
-I
&I ..
(. (t. - r,l / '!"c J
I
!unction of time.
Fig. 13, Photopaph of parallel pb,te model teat, with gnld leal' Hakell tracing ~Iinea : left aide li&bt auid, ri,bt Bide heavy fluid; exposure tim e (1ated in tbat study "'ere all !ingle-valued. There are, however, eases of practical importance where the hodogrnph is ",,,-ny_valued. An enmplc is a drain age "'ell 10eated in the vicinity of the l\I1l1COast to recover part of t he fJ'l!:'lh water t.ht floW!! out to the sen. In their progress report Bear and Dagan [ 1062] suggested the po&'libility of many-valuedness of the hodoglltpb in that case. This dminage problem will be treated bere to show how to deal with hodographs wben they are many'·allled. Many-valuedncss occurs when the s:tme specific dischrge vector is encountered in se"eral different places of the aquifer. PolubarinooaKochina [1952J shows tbe shape of a double"alued Itodograph in the case of seepage through '" dam, without (unher mathematical eltobora_ tion (see 1962 traHalation, p. 46). To 8how the characte r of II double-valued hodograph finn in a sim plified form, the one fluid C8!1t: of ...·ate r flowing arolmd a .. ~1I while a drain interrepis part of that water will be treated hem (Figure 10). Double-valuedness is a ""nsequence of t.he inflection points of the streamlines in the region CSGPD. The two-Huid cue of fresh waler flowing Over a stationuy salt wale r region with 11 drain
shO"'! a !imilar pattern of the streamlines (BCtl Figure 2m. All Sgum. are drawn {or the cue II = e, Ow= 2Q,.
192
Soil Mechanics and Transport in Porous Media
545
A i\1wI !I- Vulll(d J/ udogrupll 111(>1l
be written as
in the colllplex ,~ pinH". Hero w is defined as
1O = u+w "' q.-iq. By this definition the relation between
- or,
'" J' ~ "" J' >.,,,+a,
(A_a) ~ +fJ
2a I'" I
til
+ a, (" a'" - ,)
To obtain the complex specific disdmrge is applied to (1)
• Q,
10 = -\ ..- (a
1m (nCt)] ... -Q,
Re (I)
for
at permits
G:
n(I)'" 0
ev~lull.tion
>
for
a
1m (I) -
0
t = 0
of the constanta
:o d::ro/21TT
(34)
where T is the distance from the point of vorticity. Written in terms of the streamfunction IJI, this is
which by integration gives (35)
The constant is not a function of 0, because qr = 0 since t he discharge only rotates around the point of vorticity. Thus, 81J1180 = O. Since we are only interested in the derivative of IJI, the constant is irrelevant and can be taken as zero without loss of generality.
Selected Works of G. de Josselin de Jong
217
9.
GENERATING FUNCTIONS
391
Because all t he regions containing vorticity contribute to 'P in this manner, the total expression for 'P as created by all the vortices is the integral (36)
where In the multiple flu id case where density variations create vorticity according to (33), the expression fo r 'P becomes lJ'1(X, z) = ( - k/hp.)
ff [O(pg)/ox](Iln r dX(I dZ(I
(38)
T he surface integral for 'PI covers the entire region between the boundaries containing fluids. This expression for 'PI forms t he vortex part of the streamfunction. I n general, the values of IJI, on the boundaries differ from the values required by the boundary conditions. These differences form a new boundary value problem to be satisfied by a harmonic streamfunction IJI II obeying the Laplace equation V21J111 = O. The function 'Pn forms the second part of the solution. T he fi nal solution to t he boundary value problem is IJI = IJI I + 1f'1I . T his way of solving the problem is known in potential t heory as the solution of Poisson's equation by singularities. I n this case t he Poisson equation fo r IJI) is (16) which has (38) as its solution. In a similar way, a solution fo r the pressure p or the multiple fluid potential 9 can be given because they obey similar Poisson equations i.e., (17) and (23). This gives PI
= (- 1/271-)
fJ[o(pg)/oz](Iln r dxo dzo
9 1 = (+ k/bTP.) JJ [z V2(pg)
+ O(pg)/oz]o In r dxo dzo
(39) (40)
Also in these cases, add itional functions Pn and 8" obeying Laplace's equation must be introduced to complete the solution and to satisfy all boundary conditions. An alternative way of decomposi ng the solution of Poisson's equation fo r boundary conditions is pointed out by De W iest (1969) . I nstead of In r which can be considered as a Green's funct ion for the infinite domain, Greens functions can be introduced which have zero conditions
218
Soil Mechanics and Transport in Porous Media
392
G. DE ]OSSELIN DE JONG
along t he boundaries of the domain considered in each specific problem. (The function or its normal derivative is zero along the boundary.) The differences wit h t he required bou ndary values are then those boundary values themselves, creating a boundary value problem with a solution satisfying Laplace's equation. Since, however, a boundary value problem has to be solved anyway, the method proposed here seems less laborious because it circumvents the construction of a Green's function, which in contradistinction to In r is different for every bou ndary geometry and for every point in the field. We are now ready to compare the torque, caused by density variation in a circular area as determined in Section 4, to t he driving force necessary for the circu lating discharge caused by the vorticity. If, however, the vorticity is limited to the circu lar area, the available torque is not sufficient to support the discharge outside the area. This outside discharge requires a torque of infinite magnitude. T his discrepancy is a consequence of the vorticity, being caused by density variations. In other instances vortices exist which can be limited to isolated regions. In this case it is impossible to visualize an isolated region containing a density gradient. Isolation means that it is surrounded by a field which, up to infinity, has the same density. The grad ient actually means that the field contains fl uids of different density. If th is difference exists up to infinity, this entails also a torque which is infinitely large. In order to circumvent this difficulty and to obtain a realistic situation, the case of a circular aquifer of finite dimensions an d surrounded by an impermeable boundary will be considered (see Fig. 4). The radius is ' 3, the thicknessJ. and t he porosity n. For the analysis to remain simple the density will be taken to vary linearly over t he field accord ing to (41)
where Po' is a constant. For this situation, (aplax) is constant over the fiel d and equal to 8p 8p 8~ , 8x =8g· 8x = Po
COS(I
(42)
From (33), the vorticity is
0 < , < 'a
Selected Works of G. de Josselin de Jong
219
9.
GENERATING FUNCTIONS
393
p
~
'~ .
. F! c. 4.
which is also constant. As explained above, an integration of the influence accord ing to (32) of all the vortices in the entire region gives the solution similar to (30) 9s
=
wr = - !(klp)[8(pg)j8.t] r
(43)
Since this gives directly a discharge perpendicular to the radius, this solution satisfies the boundary condi tions which have actually been chosen in this example in such a way that the discharge obeys the simple formula (43). Therefore, the second part of the solution is zero. Since in Fig. 3, qs is positive for counter clockwise rotation, (43) gives a clockwise rotation for positive p~. I n order to show that the clockwise torque (7) is apt to create this circulating flow, the torque will be computed here by considering the required forces.
220
Soil Mechanics and Transport in Porous Media
394
G.
DE JOSSEL I N DE JONG
T he force necessary to drive fluid t hrough the ring between rand r + dr is
Hence the total torque clockwise in the circular region is
(44)
A com parison wit h (7) shows t hat if r = r3 , the two torques are equal. From this analysis it follows t hat the rotational flow within t he circular area is possible because a torque created by density variations acts on the flui d of th is region. This is comparable with t he general concept t hat forces of the boundaries cannot create rotations in the interior of t he fl uid . Vorticity can only occur in those places where body forces act in t he form of torques on the fl uid. The geometry of Fig. 4 gives a fl ow pattern, which by its simplicity permits us to predict t he rotation of the fl uid at all subsequent t imes. The discharge is such that t he Au id rotates as a rigid body. Therefore, regardless of dispersion and diffusion, t he Auid conserves its density distribution (41 ) if t he coordinate g is aHowed to rotate clockwise with t he fl uid. This means that the angular velocity is equal to
Integration gives fo r
01
=
0, t = 0, (45)
7. Multiple Fluid Potential for Variable 1-'. P. and k Equ~ t1 on ( 1,6) I ~ no t co rre et.
Sl neo tY h
e· ~~ "
-+
defi ned by (1 8 ) t o be
~~ E z
"
(46)
.. 0 lin d with ( 21)
(47)
Selected Works of G. de Josselin de Jong
221
9.
395
GENERATING FUNCT IO NS
ther efore the contin.,it)' equation
~ ho"ld
be
(48) It is l.mpo"s1bl" to b uild " P"i~non equation tn
e
f ro," thh ""pfe n t on and
t herefore .. l.. it 15 impos,dble to c'Ons~ruct an 1nte (, r~1 e~prcsRlon for@ ·"f the f or,"
(~e).
which represents the rotational part of the solution. For 'P and p, the equations are
~ B
..0.," .0. •• ,, =. 0"'" ••• • .0• .," • ,• "'" •
application of silrlilar transformation formulas as (2.3) giving
0
0
• 0" 0 • • • 0 0, • "
0
0
0
• 0
and this results Zfl - 0
0 .0'
d "
0
0
0
cos 8
•
sin
0
}12 "
0
wIth (Z.O) in
,
20 -
:lOs- 0 ;
(KIll)
(2 . 8)
Vy sin o.
ThiS r esu lt indicates . that the vorticity in a point created by a s pecific weight gradient in the pore fluid. has its a d s
t..lnqel'lt. to tl\e intu5edlon ]inc
of a horizontal plane and the plane of constant y. This vorticity is represented by the d oub l e poin ted arrOW in fig. 3. Its
mo.l.e of rotation iB
such as to r e v olve the y _consta nt plane towards a horizontal position . a physically pla USible action .
Sharp interface A sharp interface between two fluids . of specific veiqht
n.
1'2 respectively.
can be considered to be a zone of thickness h of gradually decreasin9 l' fro .. 1'\ to 1'2. in the li..it that h is reduced to zero . The zone contains sheets o f planes for y _con stant, that a re locally parallel and in the li.. i t are sqeezed toqether i n to o n e pla ne. Consid er a n infinitesimal area dA of the interface . This area corresponds to a volume hdA of the zone of thickness . h. In this volume the vorti c i ty is accord i ng to (2.8) and the tota l strength o f the vorte" actinq is 2fl
234
t
dV -
(KIll)
V'( sin
(1
hilA
(2 . 9)
Soil Mechanics and Transport in Porous Media
4 (1979) DELfT PROGRESS REPORT Since y is a",sUllled to deereaSe gra dually within the zone we ha v e
"
(2 . 10)
and 80 the vortex acting in an area dA of a sharp interface has the magnitude (2. II) The direct-ion of the vor tex corresponds to the positiv e t-coordinate, which i. locally tangent to the line of intersection between planes of constant y and the horizontal plane. In this case the planes of constant y form a staCk, that is sqeezed into t.he "'harp int.erface plan". So the vortex ha s its axis parallel to the horizontal contour lines of the interface plane. A sharp interfa c .. plan .. bet.ween t.wo fluids of different density actually is a vortex sh .. et. . The horizontal contour lines are the vortex line .. On that .. heet fi9. 4. The ribbon cut out by t.wo adjacent. horizontal c ontour lin .... form" a reentrant vortex tube . Let. the height. distance bet.ween the contour line s be Az. Then 6z is an constant for these two contour lines and the width of the ribbon
is
~/sin iang 14 abc on
inter/ace with local n, 6, t coordinates . P if! projected on the s , t plane in point P' .
Selected Works of G. de Josselin de Jong
239
GEOSCIENCES
99
When the distance Ro bet.... en the centre of gravity of a triangle and the point P is larger than 2S tlloes the smallest triangle side, an errOr s",,",ller than 0,02\ is introduced by concentrating vorticity
ot
the triangle in its
cent.re of gravit.y . 'l1>en the int"",ral. of (4.2) slloplify. because s· . 0 and Ro. The result is
R ·
, ,
(-s )
1""
(+n )
J
with A - !ldS' dt' _
the area o f the triangle. However, if P is closer to the
triangle, lnteqration has to be performed with s'and t'v&riable. This MIOunts to & cumbersome analysis with the following re s ult.
Re8Ult of integration OlJer triangl€ Let the cornerpoint s a . b, c of the triangle be located Such, that viewed frOlll above the tri&ngle is at the left, when tr,,"versing the sides from a to b , b to c. c to a. see fig. 6. Let. P' be t.he project.ion of the point P of consideration on the s . t plane (the plane of the triangle). The distance frOlll P to P' is equal to np the coordinate of P normal to the s, t plane, np is positive for P above the plane, negative below. The projection of P' on the Hne ij 18 P
. 1j The specific dischar ge components qn qs created by the triangle abe then are
(4. S)
(4 .6)
Summat i on is over the sides of the triangle, such t.hat 1j are subsequent.ly ab, be, ca . The line pieces bet.ween
I I
are always positive. The line iPi:l
is postive, if the direction of going frO P
to i is in the arrow directions ij as indicated in fig. 6. In fig. 6 the line aP 1s therefore negative. ab The line P ' P is positve,if P' 1s Located to t.he left whe n going from 1j towards j. In fig. 6, P ' P is positive, but P ' P is negat.ive. The factor f ab be equals 1, if all P ' P are posit.ive. Then P' is located within tile triangle ij abe. The factor f is l-ero . it one of the lines P'P is negative. Then P ' is ij located outside the tr iang le. The factol' f is introduced in order to limit the arc t an () to values between -~" and + ~" .
240
Soil Mechanics and Transport in Porous Media
100
4 (1979) DELFT PROGRESS REPORT
SingwZar behaviour in a nodaZ point The expr es sion (4. 5) becane's i nfinite , when the point P approaches One of the sides of the trianql~, because either """,,,rator 0 " denomiMotor in the loqarit~ becomes ~ero,when p and P conincid e. This in ter fe r es unacceptably ij with the determination of the speci fic disc~rge in a noda l point. since all
a d jacen t t"tangles then contribute infinite normal components . In order to establish the influenc e of these tria",:!L"s .. specia l procedure has to be
f ollowed, based on
II.
detailed &nalys i s of the s pecific discM rge s encountered
in the vicinity of the interface. At a noda l point severa l tria ngl e s meet, in " diamondshllped interface
schematisation . Therefore the substi tut i on by tri&n91es is not appropriate a t " nodal point . because instead of the top of " pyr.u>i above or below,respectively , !le
t ---"\\
r
Fig . 7, CiI'cuw pW't of inte,.fs ,
Specific diBCharoge in P on n01'mlete Edel""'" interf""e . Ilis treatis" st a rts with a she"," nO\l conditions ",,01 the Di~t~_Dupuit clari fying d"",cription of the "",ch81>i~m of .. pproxi .... tions. A solution is verified with ground"o.ter now ",,01 its relation to pres_ " ,..,ault fro .. exact vortex t heory. sures , expressed in terms of e'l.uilibrium o f forces. From pase 642 on , hO'JCver , the INTRODUCTION analysis is contused by the presupposi t ion, that groundw&ter flow i. alwa,ys s~je ct to \>0. Shear flow tenti .. IR and the treatllllnt continues wi th ~b e n tvo fluids (1 , 2) o~ di fferent spe d fic (J. fo r'" of Do.rcy'a 1 .... , t." .. t is uncapable to d~8cribe variable density now , correctly . weight rl' Y2 resp ectively, occupying .. n Applying ...~ appa.rently rigourous r easoning &one nts 'I. I ' 'I.n2 in shown , that rotation e xist . propOrtior.a.l to the two fluids at eithe r side ~t be "'1. ....1 th e hor;'.ootal component of the density gra_ in order to satisry continuity . In forl:lula dient and hO\l the sped ric diseh..,.,.., can be corcputed ;n th e {::tire "'lui fer by locat ing ( 1.3) vortices in the r egion" of rot .. tion . By this theor)' it is pOsdl.l e to tre,,' any
75
244
distriblltion ot densiti e~ , abo ot tit" gra.d""l tr",,~i tioo U>I'Ies , th!l.t exi~t beween miscible rlll;d. beca ... e o f difrusioo ..... d di s persion , "n>e theory i. veriried in the pap"rs by redlleing a !l:radual tr ...... ition ~one to • sharp intert..:e ard ""to-blishi"g the ..... gnitude ot th e shearfl"" , 'lhe &olu_ t ion~ then become .; nglll"r .:,d proving a .. inac cu"""y mentioned by Dietz (1953) PI> 88 . "n>i. lead. to Bear'. e'l..(9.5.6~) which i. similar to Dieu ' a (2 1,a) ettept for a com_ mutation of a nUlli>er of ter ms with the Second ./ax oC .'n/3x· , eo..,,,riaor. of the tim pape,.. is so"",vh,,t i",eded, bec",,,,e in a.ear ' s (9 .5.6b) the nu""rator beween bra""" is mlspr.inted . The reader c"n rea.-eoo.. din .. t.e i8 " e rti " .. l, th e x,y coo,,_ din .. te a ....e h ori WIlt. .. I . 'nl e hei gilt 0 r th e interr"" .. i . c ...lIed l; . So UI .. eooenu at the interfac e . The norral .peci fie disch!Ll"ge eO"floncnt f ... rd~ with a "'lo~ity ~~/at, a surfac e of a""", A ot the interfa"", plane ..,""" durin!'; '" ti"", inter_ val dt , UlrouW! a vol~ o.f the aquifer o.f ""'l!7'itude c08a(3~!3t)dt . A . (aee fi!,; . 2) . 'llle ",,,,,,unt of fluid in thi s &qui~rvo l urne is c times the volume, wh"n C is t.he ..,ro._ sit;( Of the aquifer . So. the a"",unt o.f fluid displac e d is ec""a(3~/h)dt . A . The speci fic discharge 'In i$ det'i.ned "" the w1"",, of nuid p""sing IIor"",l t.o t.lLc iat." .. Cace . throu;.i' .. u , it. ",.cs d urin r. u"it ti ... . So. the a""'\lOt cf fluid p&.'!sing th~ !.he ar"a A dur in!'; a ti ... dt i s '!nAdt. CCqlaring the w o vcluroo" gives ~", £ccSCl(3~/3t)
Fi!,; . 2 _ I n tA! rfaee "ovin!,; upwardS
3 n.TERrACE )OTICU EQUATION
Continuity in x -d irection
Let th e a quifer have iu upperboundary at z-Z) and its lQller boundary at z-Z2' '!h e heigllt occupied by fluid 1 is (2 ,-~) and for fluid 2 it is (I; - Z:!). s e e fig. 3 . 'llle hori ""ntal .speci ric dischar~ components 4 x 1, 4,&""ter . ¥/he" t he viscosi t ie. are equal , su:;h that ~)"'l-'2"'~ . eQ.uatiOf1S(2 . 16) , (2 . n) redu:;", to
. (Y2- Y, )Slnoc-csa (q", -IOti"" ,.quatiOl1 . eo ... parable with Bear ' . (9 .7. 20) . \/hen the lo_ ca ti on o f U\e interr., .. i . Itn""", aL • p.rti cular ..... n t" il Itn (Jlm • • • tuneti"", or :r. Ntd the "P'.rds .ation 3r;13t o f th .. inter_ fac .. e ... W dt'terllined with (;3.12) for e"",,.,. point o f it . In the Cue of t"",h, aalt groundw.ter tbe ditference bet.... en II I ..~d ~ ;. . .. ll and introd "dn, 111"')12'"11 gives
(3 .
,~ )
In the axid .y ..... trie eue , "ith .. the ho r-'ial coor1iinate, th e inl ern..,e IDltion equation , v ith S,-5 i, 2 ri~lal
-o ,
((a~nt).(Q/211rlf; 1 (Z' ''2.:! -2~1/2(Zl -Zz )j . • ' { (Zl -I;)(~-~)(a~/a r ) ) .. ~ ar r (Z _z H l e..., .nd thoBe ~veloped in liUrat"..., is in the t e r., eon taining ""isotropy: ("III" .. ) .
" 248
Soil Mechanics and Transport in Porous Media
Ch&r&:)",, , that the resu lt (B . 3) gives the s""'" values as (k .91, (4 . 10) Md (4 . 13) , (4 . 14) , nuicls through porous me
mean velocity of liquid How
~.
l~
"" g~
distributiQJI
po"" P probability function
lunttiQ!I~
for choke of
X, Y, Z, R coordina tes of arrival point of a T
p&rticie, starting in the origin time of travel of the particle
N
X"
number of canals covered by the particle y., Z . , Reo coordinates of the maximum
number of particles traveling during the time T. T , arrival time of the ma:rimum number 01 particles covering II. distance Z., R. N. number of eanals oovcred by the maximum number of particles ,,~,
i:illo- A I.,...,ian particle carried by the t;Urren t through the canals has to ehoose a new dire/2T
(OJ
where IT is the standard deviation of Ihe probability distribution. As P(X, Y , Z , T ) is a function of X, Y, Z, and T we may dcri"e for a gi"en T, the place X, Y , Z where the probability is a manmum, and the standard deviation ax, IT" a z in the directions perpendicular and paranel to the principal .tream around this mnimum. But also we can take a fixed point in space and determine the standard deviation IT-r dcso;ribing the changes of concentration in course 01 time at the moment when the mnimum concentration passes. In order to determine tbese standard de\"iations from the probability described by (6), se\"eral integrations h\"e to be executed. I lIIegri,'!m oj I/u aprtuirm JIlT Ihe probabililyFirst the value 01 AA' has to be determined, which may be effected in II. ",II.y similar to that shown by Chandra.ukha, 11943). We encounter here, howe\"er, the difficulty that 1/ becomes infinite for 9 _ r/2 acrording to ( I) . We will not enter into the mathematical details, which can be obtained by request to the author, but infer, that 10 the neglect of small terms for N» I, the following result is obtainl-d.
264
+ Jr + /"'111
(9)
with Q.0
II'"
[In (;flu) -
_
h - ll(lI:) - (I)(s)J - - ul/6
+ (J/M
ll(") - (_)') -
c -
. . . (0) _
2i/3
(.>:) -I,y)-(, ) - O / -
t(zO) ..
tU-) ..
j{rl) -
fJja
where ( ) indicates the mean value. I ntroduction 01 tha value lor ..lA' into (6) for PI< sh~ that nut in tegrations vcrsus or, \, ~, and 'I ha"e to be executed. Wi th ""i'«t to \, ~ and 'I this is readily performed by the u"" of the well known result
L~ up (i..., -
~I ~' up (-"'/411')
,.'fJ') d" -
(10)
For the f, 'I part of the integral thill gives, using polar coordinates, a result which yields the probability, that the partide should arrive in the ring between Rand R + dR Cl~R dRlbN/)
upl -R'/4NJ/
(It)
From the,., I pa rt of the integral only the integra_ tion wrsus \ may be treated according to (10) aiving (2 ..).... dT dZ(r/,,\'j1
1~ d.
+ i{bM.lz -
X
t:. the root of N .. j .. ~/(~-t-In,..)
(Il)
T he relationship between;>' and N according to ( 13) is represented in Figure 2. Again the larger N, the bettcr the approximation. With Ihis substitution, the coefficient of,' in the integrant is independent of ,., and (10) may be used. Then finally the following resull ill obtained for PN •
Soil Mechanics and Transport in Porous Media
D I FFUSIO~
,
From (1(;) il is ob,·ious that the standard deviat;nn of P(JI) approaches the value
•
•• - 1-
r"
/
'"
/
•
_
• •
•
••
•\
, . "1"11'1, ,.. .. IOV Z./ J
FIG. 2 - Graphico.l ~pr_ntation 01 Eq . 13 combillCd with F.q. 21; ~ as a lunction 0174 1b~ diSlance of the maximum """«nlrati",,. and f Ihe eI~m"ntar)' canallenglh Z, R) dT dZ dH - (dT dZ HdH / S.-N'j(a< - b')1)
Xe>q>I - (A+A ')N- '+8-CNj
(14)
,,·herein
.z _
leT' - 2bTZ
B _ 21ar - be,T
c _
PI!') d" -
1«1' -
2~
d.
C'
."
L~ explFl" J
,
(I i)
exp
WW!
dl'
c.,n by ,·;rtue of the substitution (16) be 31'· proximatl~l by
•
f--/ , , •
F"(}..JII
The evaluation oi an integral of the lonn
/
•
P.~( T,
7\
I:': GRAN UL\ I/. DEPOSITS
+ aZ'l/4(a< - ~ + dZ) + ~'ZJ/4(a.:
+ ....1/4(..,;
)
- b') (15)
- b')
A' _ R'/4J
-
U- ,...(...l](p. - ,..,)']
IZ ~/
- f··(..,))t up If·('''H
N . _ I(A
(18)
+
.~ '}/C)1
• - INo/C)1
'J
+C A')'
dTdZRdR [ PIT. Z. H) dT dX dH - 8}1,.{a< b'»)i (A
X exp 18 -1[1
+ C(A + A')~J
Fv..l
+ tv. -
~ ,)'fI"r...)
(16)
where jJ.. is the value of jJ. for whi(h F( /Io ) has its maximum. Be. can be read from Figure 2 as a function of N . or computed by (13). The standard deviation lem! in ph}'.i", and ulTOnonr, 1If;!:. Jfod. I'I,y, .• 16. 1-89, 19-43. DANel< WEus. P. V., Continuous ~ow system s; dist.i. hUlion 01 ...i,lence ,im.,., Cile"" E",. Sci., 2, t-13, 1953. 0",·, P. R .• Di.p".";on of a mo"ing saIl water boundary, TraM. A m". Grophy,. (h,;".., 37, 595-601, 1'156. [IE JOSSELIN oE JO,,-C. G .. l.·ento:rtainemcnt de l>arlicui ... par It courant inle.,;t ici.l. I'ubl. 41 . ,\ ssn , Hydro!,,!!)", VGGI. Sr",~,;a DdTC,!, 139-147, 1956 JAHNKE, E., A"-" F. E""F., Tahl~, Df lum/;m... Do,'e' l'ubl,. I'ew York. 382 pp .. 19-;5. KLl"KKSB ';IlC. ,\... ASO F. SJESIT1.U. H o\din~ time di'lribulions of lite GauS6ian Iy~. Che"'. Eng. Sd., ~. 258-270. 1956. K .".. ERS, 11., ,""" G !"UHD." Frequency response analy.i, of conlinuou. flo\\" ')"~I.m •. Ch.",. E"I. Sd.,
2, 173- 181,1953. RI TAI. M_ N. E., \\. J. KAUI>'''''-' "~.,, D. K. TODD Dist>~r-sion phenomena in laminar flo\\" through [>O.ous media. 1''"8'U' lI~p. l, CoM! St
H. ,\.
displacement in capittari~s. S n·ilh t .. t\ition'IIMtr .. hich dispf:roion of a lOlul 0, t
'Y = 1'1
~
'Y = 'h
r O. Thus, all the curves in Fig. 4b, which represent the distribution of qt over the height t at different time t > 0, have the intersection with the axis t = 0 in commo n. The heavy curve in F ig. 4a in tersects the axis w = a in ro = -0.503. This means according to eqns. (17) and (19) that the plane where the discharge qt is zero, is located in to = - 0.503 (O:Tqt j n)1 / 2. As a consequence this plane descends wit h time proportional to t 1 / 2 as sho wn in Fig. 4c. The curves in Fig. 4b all have the same shape as t he heavy curve in Fig. 4a but are stretched with a factor, that is proportional to t 1/2 .
\
\
\
, '
- \ i
\ \ ... '.
® Fig. 4. Relation between a solution w(r) in (a) and the specific discharge qt (~, t)in (b). The plane, where qt = 0 descends in course of lime according to ..;t, see (c). The curves are for the case m ::::: 0, f3 = 1/4.
Selected Works of G. de Josselin de Jong
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63
AUXI LIARY PROBLEM Q
The problem P(m. tJ) is not explicitly solvable, because of its no nlinear character , although much is known about its solutions. A special d ifficulty mentioned already is the occurrence of the absolute value [w i. which causes solutions to consist of combinations of parts, where w > 0 and w < O. This difficulty is solved by considering first the auxiliary problem Q, which is d efined by the differential equation: !s(du lds)
+ d[u(du lds) l /ds
(24)
= 0
with boundary condition:
u
=
1
(25)
and the additional condition:
o ~ u(s) .;;;; 1
(0
the boundary conditions (eqn. 21) show that either w(co) >0 and w(- 00) ~ 0 (case 1) or w(oo) ~ 0 and w( - 00) < 0 (case 2). In both cases the corresponding solutio n of P(rn, tl) does not change sign on the entire interval (- 00, (0). It will be shown that in both cases a solution of P(rn, tl) can be obtained from a solution of problem Q (i.e . a curve from Fig. 5 for that matter) by applying an elementary transformation involving rescaling and displacing the curve in u-direction. When jpl 0 (case 3). Consequently, the solution w changes sign on the interval (- 00, (0). This introduces an additional difficulty caused by the absolute value of w in the differential equation (20). Let TO be the value of r where the solution vanishes: i.e. w(ro) = 0 , then w(r) < 0 for r < ro and w(r) > 0 for r > ro. The solution thus consists of two parts (w > 0 and w < 0) . Each part is an appropriate transformed solution of problem Q. They are joined together at ro using the continuity of the concentration (or velocity qt) and the continuity of the saJt flux. T he treatment here is aimed to describe the required procedure of rescaJing and combining the curves appropriately. without too much mathematical details. For more detailed information the reader is referred to Van Duijn (1986b).
t
Casel:{3 ~ !
When (3 is larger than ! , both fresh- and salt water flow upwards. When tl equals!, only the freshwater flows upwards while the salt water is stagnant. Both situations are shown in Fig. 2. Now let u( s) be a solution of problem Q and consider for 0 > 0 the transformation:
r = os
(32)
w(r) = o2u(rl o) - m
(33)
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283
67
This transformation consists of a rescaling (caused by 0) and a displacement (over m) of a relevant curve from Fig. 5. By the transformation t he rliffp.rp.ntial p,quation (24)
tr dw l dr
hecomp.~:
+ d[(m + w) dw /dr] / dr
= 0
(34)
because the a cancels. Next let u(s) be a solution of problem Q of type I or the separatrix. Then u(oo) = 1 and u(- 00) = ~ ;;a. O. Using this in eqn. (33) gives: w(oo) =
02
-m
(35)
and: w(-oo) =
0 2 1/1
- m
(36)
Thus when choosing a such that:
(3+t = o2 - m
or
o = «(3+!+m)l fl
(37)
and O. From eqns. (37)
284
Soil Mechanics and Transport in Porous Media
68 1. 5 ... .
'l' ~ /q
...... __ ... .......... ........... ,,-
1·'SI •
"
'"
., ... .•.... ......... . _. _. _ · JJI
_• .0
,
-2 ·0
,.,
•
Fig. 8. Similarity solution for m = 1, (j = 0.881.
and (38) it follows that: 0 = 1
.nd
(39)
" ~ 0
As mentioned ahove, this value of.p indicates that the relevant curve is the
separatrix of Fig. 5 and the value of a shows, that the similarity solution is the undefonned separatrix. Since this simple result produces a clarifying example, the corresponding specific weight and specific discharge distributions are shown in Fig. 9. This figure is to be interpreted in the manner as explained for Figs. 3 and 4. In Fig. 5 the separatrix is specified by the intersections with the coordinate axes. These are: s = 0 So
= - 1.23675
uo = 0.5873
u
=
0
I
(40)
These values are reencountered in Fig. 9 in the following manner. Since the scale factor in this case is a = 1, see eqn. (39), the s, u values are directly the r, w values, which are related to physical quantities by eqns. (17) and (19). Using these relations it follows that s = 0 corresponds to t = 0 for all time t which is the original height of the fresh-sa lt interface. The first line of eqn. (40) therefore indicates that the specific discharge at the original interface height is constant for all t and equal to qt = 0.5873 q, see Fig. 9b. The second line of eqn. (40) indicates that the depth to, below which the groundwater is still stationary and the specific weight is not yet reduced, equals - 1.23675 (aT9t l n) !12. This means that this depth increases proportional to root time, see Fig. 9c. Case2: (3';;;;;-~
When (3 is smaller than - ~, both fresh- and salt water move downwards. When (3 equals - L only the salt water flows downward while the freshwater
Selected Works of G. de Josselin de Jong
285
69
,
©
Fig. 9. Example t-ii, case m = 0, /3 = 1/ 2. Development of brackish zone, when molecular diffusion can be disregarded a nd the sal t water is statio nar y.
is stagnant. Both situations are shown in Fig. 2. Now the appropriate transfonnation is :
r
=
-as
(41)
- w(r) = Olu(- rio) - m
( 42)
Applying this transformation, eqn. (24) becomes: irdw/dr+d{(m - w)dw /dr] /dr = 0
(43)
Let again u(s) be of type I or the separatrix. Then eqns. (41) and (42) imply that : - w(- oo ) =
0 2
(44)
- m
and: -w(+oo) = o2¢ - m
( 45)
Now a must be chosen such that:
-{3+! = o2 - m
a = (-
tJ + ! + m)1 12
(46)
and ¢ such that:
-,B - !
= o2¢ - m
0'
'"
~ ( - ~ - l +m) / (-~+l +m)
( 47)
in order that w(r) satisfies eqn. (21). Moreover, w(r) '" 0, implies t hat -w(r) = Iw(rll. Thus eqn. (43) is identical to eqn. (20), showing that w(r) in this case is in fact a solution of P(rn, ~). Summarizing, for this case t he solution of P(rn ,~) again consists of a type lor separatrix curve from Fig. 5. Since both eqns. (41) and (42) contain a minus sign the relevant curve from Fig. 5 is rotated over 1800 to produce the similarity solution in the r, w plane.
286
Soil Mechanics and Transport in Porous Media
70 Case
3: -
t < (J < + t
In this case fresh- and salt water flow in opposite directions (see Fig. 2) and so q~ and therefore also w from eqn. (19) change sign in the region of integration. The positive part of the solution is denoted by w· and its
negative part by W - . Then w+ resembles case 1 and w- resembles case 2. Before showing the practical elaboration of two examples in the sub· section "use of Fig. 10" below, a few concepts required in the procedure are mentioned here first. The positive part of w(r) is defined according to: ( 48)
with
and the negative part by: ( 49)
with
In eqns. (48) and (49), u· and u- are parts of two different curves from Fig. 5. Because of the minus signs in eqn. (49), the part u - is rotated over 1800 in the rescaling process. By choosing 0 according to eqns. (48) and (49), it follows that w+(oo) = (3 + , and w -(~ 00) = (3 ~ ,. It remains to organize the solution in such a manner that the two parts w+ and w- match together at the point where w = O. More precisely, it remains to select ro and curves u+(s) and u-(s) from Fig. 5 so that the composite function: w+(r)
fm
r> ro
w(r) =
(50)
r 0 the fluids become more or less mixed and the specific weight 'Y becomes a function of the local height and the time t. The specific discharge qt changes accordingly because it depends linearly on 'Y, see eqn. (11). Because of the plane fl ow and other simplifying assumptions, the system of partial differential equations, that describes the spreading process can be reduced to the single differential equation (20). This is achieved by introducing the similarity variables wand r. From these, w is related to the specific discharge qt by eqn. (19) and to t he specific weight 'Y by using eqn. (11 ). The variable r is related to the height and the time t by eqn. (17). The governing equation (20) has as relevant solutions the family of curves
r
r
r
290
Soil Mechanics and Transport in Porous Media
74
shown in Fig. 5. By rescaling, displacing and combining in various manners t he appropriately chosen curves of this family, it is possible to construct t he solution for various values o f molecular diffusion and flow conditions at infinity. Molecular diffusion in comparison to dispersion is described by t he pnrrunet.cr m, see ..qn. (115). the flo w conditio ns o riginally and at infinity by
the parameter (3, see eqos. (4) and (7). The differential equation (20) is a nonlinear diffusion equation with (m + Iwl) as diffusion coefficient. The absolute value of w in this coefficient is unusual and requires a special treatment when w changes sign in the integration interval. By eqn. (1 9), this o ccurs when the fluids flow in
opposite directions. In this paper three cases are considered that differ in the way the two unmixed fluids flow. In the cases 1 and 2 both fluids flow initially and at infinity in the same direction . In case I, the choice fJ ~ i guarantees that both fluids flow upwards or only the freshwater flows u pwards and the salt water is stagnant (qt ~ 0). In case 2 «(3 '" - !). both flow downwards or only t he salt water flows downwards and the freshwater is stagnant (q ~ '" 0). In case 3 (- i uj(.o). Then two situationacan arise: (1 ) There exists an other intersection point 510 such that:
,.,
,.d (2) UI('»
111(') for ali I > '0 and both solutiona sat isfy eqn. (25).
Ad {lJ
Integnlion of eqn. (24) with respec t to. from '0 to '1 gives for II I and ul, respectively: Uj('j)u'I{'I) - III{'O)U'I{'O)+!'IIII(Bll -!aOIlI('O)-!
J-,
and :
'0
U1('I)U1('I) - 1I1('0)U;('0) + hIUl(SI) - houl('o)
Subtractinll" thlllie equation. lI"ives:
'I
UI('I)[U'I - U1](.d - udso)[u;-uil(so ) - !
J
-,
- !
ul(.)d, = 0
J-,
ul(.jd8
(M)
'0
[ul(') - 1I1(')]& = 0
However, by the above aS6umptionl 1 1I'I - U~]{.d O. This contradicta the equal sign in eqn.
-,
0
(A3)
(A5)
11I', - u;](.o»O and (A5). Therefore case (1)
cannot arise.
292
Soil Mechanics and Transport in Porous Media
76 Ad (2)
A similar argument gives for this ease also a contradiction. Thus both (1) and (2) cannot arise and thus no interseclion point exists.
Asymptolic behaviour Next an argument due to Peletier (1970) is used to obtain eqns. (27) and (29). The starting point is the following observat io n. At points where u > 0, eqn. (24) can be written as: (A6)
which can be integrated to give: (ul)'(s)
:=:
2 (u )'(solexp
f.,•
{- 1
(A7)
[z/u(z» ) dz}
>
Here 80 is an arbitrary chosen point, where u{s ol O. SinC(! 1.1(.)< 1, it follows from eqn. (A7), that: (u2j' (8)";(u2)' ('O)eIp( _ lt.2-8~)J
for
8 > '0
(AB)
Using 1.1'(.) > 0 and thus I/(s) > u{sol for 8 > 'a in eqn. (AS) gives: O
Selected Works of G. de Josselin de Jong
293
77
Let ~I
=; + i&, wh(!re &
from,/ to
gives:
'i.)
(U 2 )'{'I+d-{u 2 )'{.;l+
.
denotes the discretization interval. Inte gration of eqn. (81 )
' j+ I
J
su'(s)ds
(84)
0
=
1
f.
t / l.+c
n (O)A(O)CO'(O - 1X)dO.
-1 / 1.+.
Integration is performed over the range - -ill + t:r: < 0 < + -ill + 11, because only channels in those directions carry waler away from the junction points. That only discharge departing from the junction points is counted, is a require-ment for the probability calculus. Thcchoice is made forward in time and not backv..ard. The x-component of one step in the Markoff process is the projection of the channel in x-direction. For each of the n(O)dO channels, trus is IcosO. The mean value x of these x-components is this value leosO multiplied by its probability o f occurrence p(0 _0 + dO) integrated over all possible directions. This gives with (10), ( 11), (12) I / h+.
(13)
302
f x-
n(O)l cos Oq(O)dO
-1/1.+·
Q
If n(O),J.(O)cosOcos(O - ll)dO
-
f
n(O)..1.(O)cos (O - ll)dO
Soil Mechanics and Transport in Porous Media
11u! Dispersion eM/fic/tnt in An/sotropic Porous Mtdia
For the y-component, which is /slnO, this gives _
( 14)
f
IJl •• ~
-1 / 2.+0
y~
f f
n(O)IsinOq(O)dO
'"
/ n(O)).(O)sinOcos(O-Cl)dO -",--------
Q
n(O)J.(O)cos(O - Cl)dO
The residence time I of one step in the Markoff process, is the time a particle stays in one channel. If the discha rge is q(O) and the volume of the channel is / . c(O) then this residence time is I = (I . c(O)/q(O». The mean value i of the residence time is this value multiplied by its probability of occurre nce, P(O ..... 0 + dO), integrated over all possible directions. This gives with (10) ( II ), ( 12),
, f==--111.+.
- 1/ 2.+0 n(O)/·
_
(15)
I
c(8)dO
~
~
-
n(0)c(8)dO
- ' ; 0 - --
I V¢l1
Q
f f
n(O)J.(O)cos(O - Cl)dO.
In the same manner we find
= _ 1'-,f'-c"(:.-o~),,:-(o,:,),",,o_'_'_O'~O_'.:.(O_-~'~)d_O n(0)1(0) cos (0 - Cl)dO The general case of anisotropy is obtained by assigning to the n(O)d{O channels an arbitrary distribution with respect to 0 of the combined conductivities n(O»).{O)dO and the combined cross sectional areas n(O)c(O)dO. Every possible distribution can be expressed as a Fourier series. Let these be Xl
(16)
J
{
(17)
n~) ),(0) = 1:
._0Ahcos 2n(0 -
"N(O)c(O) =
..1:,
Cl2.)
Blocos2n(O-{Jl.)'
Unly the even terms appear, because every channel figures in two oppposite directions. Execution of the integrals in (13), (14), (15) and (16), all between the limits - til: + a and til: + a gives finally . I i = R
(18)
Y=
~
-
I
;
[A ocosa
+ tAl COS(2Cl 2 -
; [A osinCl
+ tA 2 sin(2a 1 -
II:
Cl)] a)]
B
'~RI·" I' with R =
f
1110+.
nCO ) - - J.(O)cos(O - a)dO.
-1 120+0
Selected Works of G. de Josselin de Jong
N
303
G. De Josulill de long
'66 The real mean velocity v, = f ll. With (l8) this is
Vx
= Xjl and
V"
=
~
=
[2~o COS 11 + 4io COS(2IX~ -IX)] -I V 1
v,
=~
=
[2~:SinO:+4~2(1Sin(20:2-a)]· IV41I.
(19) {
has x- and y-components given by
II
Since the gradient of ¢> has the direction IX, the partial derivatives of !II can be V¢lI' CQsa, ot.Uvsin. V. N . (1959). Convcdivc dilfusion in porous media. Prlkl. Mat. Mddt .• fl, J042- J05O. 3. ScH:rm.ooa. A. B. (1961). Ge:ners.l theory of dispersion in poroUI media, J. Chi)p/I. lW., _, 3273-3218. 5. FD.a.umoH. 1. (1948). Les loil ck 1'6eouJemtot de filtralion, u Gi,w Clril, 124,
>4-21.
ENolNualNO D.,......11oCJiNt'. DaLn U~ Of 1'lIcHNowo¥,
Qvn.
00rrJ'1.AHT1OI!W
25, DILl'J', N8'J'HEIll...oUo
Selected Works of G. de Josselin de Jong
305
DISPERSION IN FISSURED ROCK
by
G de Josse1in de Jong and Shao-Chih Way
Geoscience Department New Mexico Institute of Mining and. Technology, Socorro, New Mexico 87801 July 1972
306
1
Introduction It is the purpose of this study to show that the dispersion of particles, transported by a fluid flowing through a system of fissures in rock, can be treated as a special case of the general theory for dispersion, developed by use of the probability theory (DE JOSSELIN DE JONG, 1969). A case study of the dispersion of a large amount of locally injected particles will first be presented computation wise. It will be computed how the particles are partitioned at each intersection of the fissures, how the subgroups of particles are transported through the fissures and uhere the subgroups \'1ill be located at successive time intervals. The result consists of a distribution of particles at a certain time, jr , after injection, with discrete
amoun~of
particles at discrete
points. In the second place the general theory developed on the basis of probability concepts will be applied to the same case. This theory gives as a result a continuous distribution of particles, which is essentially Gaussian. In order to show that the discrete distribution obtained from the computation and the continuous distribution obtained from the probability have the same general form, a figure a (Pig.4 ) Hill be used. i'l'umerically, the blO results ~lill be compared by considering the centre of gravity of the diapersed particles and the second moments of the distribution around that centre. The case study is simplified by considering a system of two intersecting families of fissures. Let the fissure planes be parallel to Z (Pig.l), and let the conditions in all planes perpendicular to Z be identical. Thus the problem to be treated is two-dimensional and only the
Selected Works of G. de Josselin de Jong
307
2
x,y coordinates are essential. Fluid flow through the fissures proceeds parallel to the xy plane. Discharges are per unit deuth in the Z-direction. Sections l, 2, 3, 4 deal \'11 th the computational experiment. Sections 5, 6, 7, 8 deal with the theoretical prediction. The two resulting particle distributions will be compared in Section 9.
308
Soil Mechanics and Transport in Porous Media
1. Particle Behaviour in Fissures and their Intersections The behaviour of particles in passing through the individual fissures and their intersections is partly deterministic, partly random. Let us first consider, v:ha"t happens \d thin a fissure In the follo\"1in0 the di::;charges through the fissures are divided in lamellae. Each lamella carries the same amount of discharge per unit depth. Since the fluid velocity in a fissure cross section is not uniform, the velocity in the different lamellae is not the same.
'I' '
It is assumed here, that the length, of the individual fissure segments between intersections/and the average fluid velocity, \f< ' through them are kno\m. The index, ~ , refers to the particular family of fissures (chct:r.","ct~rized b:,' their direction,
length, CQ"ductivi ty , ... etc~. In the case considered here, there are two different
kinds of fissures, so f-l has the value I , Jl. , sec Fig .1. Furthermore it is assumed that thE: Hidth, Ct'- , of the fissure is small Hith respect to its length, it'-, and molecular diffusion, Pmof , large enough for the quantity .t
"
"
/0",,, Jt' to be small w~ th respect to one. Then Brmvoian motion will force a particle to visit all fluid stream lamellae within a fissure, such that the overall particle velocity corresponds to the average fluid velocity. The time, t f , that a particle will reside in a fissure segment is then equal to Cf< iff"'
t- eis found that after the coordinates (4.4)
x
JZ
T =
for the experimental values, it 40 1.~ the centre of gravity has
= zs; 93 Y.
;
~
y =
4.91 .R.
The second moment is taken in the direction of the line of spread at 60", the result ,-,ill be called al~. 0
The spread perpendicular to the line at 60 H'ill be called ~ Since there is no spread in that direction, the
Ok
second moment, a~~ is zero.
(4.5)
316
af~ = 3.90 1.2.
So ''Ie find
...
a... ~ =
0
Soil Mechanics and Transport in Porous Media
11 5. General Theory of Dispersion
baSed.
The general theory of dispersion
0)1,
U,Stl'l$ CI-lAI!P;2ASF1~HARS [j('w;,rA~'(5p.Al~~
corresponding. to the
passage through elementary conveyor units (defined in Section G) and (t> is the average of the residence time
tm
spent in an elementary conveyor unit.
X,,,,
y", , t
h•
The quantities
are the stochastic variables, subject to a
probabilistic choice, because the elementary conveyor units have different magnitude and orientation and particles ChOOS0 to enter them according to probabilistic rules. The size, shape and orientation of the ellipsoid can be computed from the variances or second moments a"", a,,:! a~~ , ~lhose magnitude according to the general theory is given by
(S.2)
'
_ <xt> _ ,
(X~>,
are the
averages of the products of the stochastic variables 7"" Yn. , tm as cor.tbined bet\veen the brackets, < >. By introducing the second moments of the stochastic variables and indicating them (5.3)
(Xt>
-
"
.. ,de
the expressions for the variances obtain the form O;c;
= -T [ ;a - -I" t t
A
/\
-I"
tt
<x;>]
"-
z
z]
The relation between the variances and the size, the shape and the direction of the ellipse is as follows (see BEAR, 1961, by 1191). The quantities (1:>.:>., components of a second rank tensor values are a"
(5.5)
,
a~L'
all
=i
0.l.t
-==
{
f
ax~,
0 11 ,
\'Those
are the principal
given by
(au.-t 0:1,)
(O;u l'
a~,)
The roots of these principal values are equal to the magnitudes of the principal axes of the ellipse, ~ and 6", according to (5.6)
6,
=;a;;
The angle, \f , bet\veen the major principal axis, and the x-axis is given by
<S.7)
318
~_.
_ ....
6,
La~¥___ ... _________ _
Soil Mechanics and Transport in Porous Media
13
6. Elementary Conveyor Units The results mentioned in Section 5 are theoretically obtained as a distribution of the probability to find a particle in a certain location at a certain time. A particle can arrive there following different paths. The analysis consists in deriving the probability for a particle to choose a certain path and to select from all possible paths only those, that end up in the desired location at the ·desired time. Mathematically this produces an integral expression with a closed solution. It does not fit in the frame\·;ork of this presentation to expose all the details of this analysis (a separate report deals with this aspect). It is, however, necessary to mention here that the analysis requires,that the particle path be subdivided into elementary steps of a special nature. In the first place the steps must be such that it is possible to determine the magnitude of the probability for a particle to choose a step of a particular kind. Ilv -Ute-- 5CW1~ j,/ftlk it is necessary that the steps are not correlated, the reason for noncorrelation is that the individual steps must be combined into a product. It is
known that according to probability theory the probability for the subsequent occurence of several events consists of the multiplication of the probabilities of the separate occurence of the individual events, only if the events are not correlated. In order to be able to apply the mentioned product, it is therefore necessary that the steps are not correlated. When-particles are transported by a fluid through a porous system, the above mentioned analysis can be applied to predict their dispersion if the paths followed by
Selected Works of G. de Josselin de Jong
319
14 the particles can be subdivided into steps the requirements imposed by the theory.
that satisfy
We will use the
form «probabilistic step» , if the steps do indeed satisfy these requirements. Physically the particle path consists of a number of traverses through pores, cavities or fissures, depending on the nature of the porous system. In general there is not necessarily a coincidence betHeen the probabilistic steps of the particle path and the traverses through individual pores, cavities or fissures. combination of them.
It can be any
In order to indicate the physical counterpart of the mathematical steps, the term«elementwy conveyor unit» is proposed here.
An elementary conveyor unit can be any
part of the porous medium traversed by transported particles. It can be one channel bet\.;een grains or any combination of channels or fissures up to complete lenses or layers of soil. The combinat:ions of connn:i. ts consti tl\tint] elementary conveyor units can be different for different directions and magnitudes of the hydraulic gradient. In order to constitute an elementary conveyor unit the combination of conduits must satisfy the four requirements listed ·beloH. These requirements guarantee that the passage through the elementary conveyor units from their entrances to their exits, coincide with the probabilistic steps.
In the following the parameter, m,
will be used to indicate the properties of an elementary conveyor unit of the mth kind. The reqpirements are
ITI
a particle, travelling from entrance to exit through an elementary conveyor unit of the mth kind, covers a distance, whose coordinate components ?C/h, a known magnitude.
320
!:lin ,
ZIn ,
have
Soil Mechanics and Transport in Porous Media
15
m
a particle, travelling from entrance to 8::i t through
an elementary conveyor unit of the mth kind, remains within the unit during a residence time, magnitude.
tIn, Hi th a knm'.'n
"fl)
the choice being made by a particle to enter the next unit after having completed its travel through a
previous unit has a probab~lity distribution, ://11 , "'hich is a known function of the~arameter, m, of the unit it I Hill enter.
BJ .
the probability distribution of choice, J/II , does not depend on the parameter, m, of the previous unit. This independence guarantees
that subsequent steps are
uncorrelated. ·Because the values of X/11, Yhl , Z", , t/1/ as chosen successively by the particle during its travel through the porous system are subject to a probabilistic choice, these quantities are called the stochastic variables. In the case considered here, there are two kinds of elementary conveyor units. Therefore m has the values 1, 2. In order that the distribution of choice represents a correctly mormalized probability it has to satisfy the condition
(6.1) The averages (x>, , , (xx>, in th~ results obtained from the
etc mentioned theory are defined as follo\\'s by sur:ll:tations of the stochastic variables multiplied by the corresponding probability of choice (;(J> , • • • • • • • • • •
ge~eral
In
(X> =
(6.2)
{
L Xln71l1 In
= L !J",7", Ito
= I..
tin ~J~
Selected Works of G. de Josselin de Jong
hi
(xx>
::=-
(X,>
=
.t
L X", dill Irt
1 X",Y,,,7,n
•••• 4!tc
321
16 If m, instead of consisting of a finite number of discrete values, is continuously distributed, then the requirement on
~/JI
(6•3)
J
is
-1"00
?-/ll till'
:=
f
-ro
In this case the averages are defined as follows by integrals over all possible values of m
<x.'l = (6.4)
322
1m
X",
?1J1
dhl
=1
=L tin ff'I? dm
In
'!1m '1/11 dm
():x> ==
(XJ>
1 x~ '}/tl din h'
=1m
•...
X".!J)II
7", (hn
~t
, , , (xx>, , ••• etc. amounts to the use of (~.3) and (7.4), (7.5) in the expressions (6.2). This gives the following result. for the averages
<x> = ;;
111.
Sin f}1I
9][) I'lz-
( fI Cz -I-
=
..
=
.1.
.
1"
(,Pz {os til.z Sin i9.x
+ J:t
.fII~
<xJ> - o:>- = J:;
./-
1I~
"
=
- ~
1\
=
<Xi> - = ) ...
(q,s)
1:
L)(f { -
0-.1,,)"/:/...1;]
* {f 1% C&< 6.1 ( 9.t - '1JL J
)t
-t f.I Cc.r~.1 + .I C.( . . 1.06'15 .>."r"
'1, J
i [I + J,:)
,.(c.
.It:-
=
~
'fo
Xj
a dispersion ellipse with
major principal axis:
6. =- ~
b', ~ .( .u; I
minor principal axis: 6~ = o. s~ direction of major principal aBs: t 6~.:l·
E).l.
=
=0
Dispersion in this case spreads the particles in two directions because
~ ~ 0
,
instead of along a line as
,~as
the case when all fissure segnents had the same length.
Selected Works of G. de Josselin de Jong
335
30
REFERENCES
J. Bear, - __
~
I'."Sl",rt
~r
sal c.
;r
I
I
I I I I
I
I
I I
--&--I
I
I
Elementary Conveyor Units, m=l and m=2.
Selected Works of G. de Josselin de Jong
341
342
I'M!. of cuh. of 3r"'~ilj (~pe.li1l1e•.t-) p.,tk
Ii1 c~ '1 iY""'la (.fW1'j )
y
Soil Mechanics and Transport in Porous Media
t/'\
//
. ~
ill
~
Paths followed by centre of gravity of particles.
NIEUW ARCHIEF VOOR WISKUN DE (4), YoU, (1985) 207-208
Cube in Tessaract An Introduction to the FOllowing Article J .H. de Boer & J. van de Craats (Section Editors)
Tessorocf (or jour-dimensional measure polytope) is the four-dime nsional analogue of cube. In the November 1966 issue of "Scientific American", Martin Gardner devoted his column " Mathematical Games" to higher dimensional polytopes ("polytope" is the general term in the sequence: point, segmen t, polygon, polyhedron, ... ). He raised the question of "finding the largest cube, that can be fitted in a unit tessaract". lllis problem was inspired by the three-dimensional phenomenon, that inside a uni t cube a square may be constructed with sides large r than unity, leading to the surprising result, that it is possible to cut a hole in a unit cube such that a cube wi th edges larger than I can pass through it (the hole must be cut in a direction perpendicular to the square). Pieter Nieuwland (1764 - 1794) found the maximum edge length for such a cube (cL Section 3 of the following article). The weaker result that a cube can be perforated in such a way that the second cube of Ihe same size may pass through the hole, is ascribed by Joh n Wallis (1616 - 1703) to Prince Rupen, Count Palatine of the Rhine (1619 - 1683) (d. O.J. Schrek: Prince Rupert's problem and its extension by Pieter Nieuwland, Scripla Malhemalica 16 (1950) pp. 73-80 and pp. 26 1-267). In 197 1, G. de Josselin de Jong (professor of Soil Mechanics, now retired from Delft University of Technology) sent a letter to Gardner containing a description of the construction of a cube with edges b in a tessaract with edges a. where b >0. He also expressed his opi nion that this cube might be the larg. est cube that can be tilled in the tessaracl. He did not have a proof of this conjecture, which he based on visual intuition only. Martin Gardner answered, that he did not feet competent to judge the meTits of this result, and he recommended to send il to a mathematical journal. To find out wether the result was wonh publishing, de Josselin de Jong asked the advice of H.S. M. Coxeter who proposed to formulate it as a problem in
343
208
J.H. de BOER & J. van de CRAATS
the American Mathematical Monthly. It appeared as problem 5886 in 1972 (vol. 79, p. 1140). No reactions were received. However. in 1974 (vol. 8 1, p. 294) the result of Section 6 of the following article was given, but without any explanation how it was obtained. Nine years later, de Josselin de Jong, still welcoming an adequate oppOrlunity for showing the geometrical aspects of the result, came in contact with one of the editors of OUT section "Recreational Mathematics". Upon seeing the material we asked him 10 publish his construction in our journal. TIle result is
the following article. We expect that many will enjoy reading it, and, of cou rse, we also hope thai some will find it a challenge to improve upon the results by constructing a larger cube, or to supply a proof that in fact the cube is the largest possible. Also the general n-dimensional case might be worth considering.
344
Soil Mechanics and Transport in Porous Media
NIEUW ARCHIEF VOOR WISKUNDE (4), VoLl, (1985) 209-217
Cube with Edges Larger than those of the Enclosing Tessaract G. de Josselin de Jong Ary Scheffersrraaf 227
2597 VT Den Haag The Nerher-Iands
I. INTRODUCTION
The purpose of this paper is to show, how inside a tessaracl (jour.dimensional measure polytope, or hyper-cube) a three-dimensional cube can be constructed with edges larger than those of the enclosing tessaract. As an example of ales-
saraet consider in euclidian four-space R4 the polytope T = {(XI,X2,X3,X4)ER 4
. . T has 16 vertices (m
· COOT d mates
I-
I I "2Qo;;;XI,X2,X3,X4"';;; + "2a).
(-+-"2a, I -+-"2a, I +"2a, I -+-"'2a», I 32 edges (0r
length a), 24 two-dimensional faces and 8 three-dimensional cells. To show thaI T contains a cube with edges larger than a, it may suffice to give the coordinates of the eight vertices of the cube (see (10) in Section 7) and leave it as an exercise to the reader to verify the claim by linear algebra. However, it might be of interest to show, how the result was found by rotating a rectangular parallelepiped around an axis, adjusting the lengths of its edges in such a manner, that its vertices are on the bounding surface of the tessaract and its edges are of equal length. The procedure results in producing the largest cube, that can be constructed by this particular rotation. Since all other rotations the author could imagine, did not produce a larger cube, it is conjectured that the maximal cube is found . This conjecture is based on visual intuition only. and still requires a rigourous proof.
345
210
G. de JOSSELIN de JONG
2. A NOTE ON THE FIGURES
lei it first be shown bow a tessaract can be represented visually as in Fig. 2. The construction of this figure is understood by starting with the oblique parallel projection of a three-dimensional cube, as in Fig. I. Two faces of the cube are undistorted and presented as two congruent squares. The four other square faces appear as oblique parallelograms. In an analogous wayan oblique parallel projection of a (four-dimensional) tessaract in three-space may be produced by presenting two boundary cells undistorted as two congruent cubes. From these the front cube is drawn with heavy lines in Fig. 2. The back cube is displaced with respect to the front cube
in an arbitrary direction backwards. Then corresponding vertices are conneeted by 8 parallel edges. The six remaining boundary cubes appear as six oblique parat1elopipeds. Figure 2 gives a plane picture of the three-dimensional projection of a tessaract. TIlls figu re should be visualized as a three-dimensional model. It is instructive to locate on this model all 8 boundary cells, which in R4 are cubes. Before treating the problem of finding a cube with edges larger than those of the enclosing tessaract, it is hclpful to consider first the problem of constructing the largest square in a cube. This problem is treated here in a circumstantial manner, unnecessary for such a trivial case. This is done in order to introduce the elements of the reasoning which, being obvious in three dimensions, can be extended by analogy to the less obvious four-dimensional case.
3. SQUARE IN CUIIE
Visual insight suggests, that the most favourable position for the square is obtained by choosing its plane through the centre M of the cube. This centre lies (Fig. 3) halfway on the line connecting the midpoints nand n' of two opposite vertical edges of the cube. Consider this line to be the axis of rotation for a plane a, that will contain the square. Note that in a cube two edges are opposite when, although being paralleL thev are not in a common boundary face.
/'
/'
/'
/'
Fig. 1.
346
Cube projected in plane of paper
Fig. 2.
Projection of tessaract in Rl, projected in plane of paper
Soil Mechanics and Transport in Porous Media
211
CUBE IN TESSARACT
The plane (I intersects the upper and lower faces of the cube by lines parallel to the diagonals in these faces. The lengths of these lines is limited by the edges of the cube. giving symmetrically located vertices for the square. Let the edges of the cube have lengths a and those of the square b. Intrer d ucing a parameter h according to Fig. 3, gives the relation mm' = b = aV2-2 h.
( I)
The upright edges of the square are located in a plane normal to the axis of rotation. This plane intersects the cube in the dotted rectangle with sides D.. and a, such that wi th Pythagoras'~ theorem
b 2 = (D..I+02.
(2)
Elimination of h gives the relation (3)
with the root b l a = 3/2V2:o:; l.0607, showing that b can be larger than u. This is Pieter Nieuwland-f*i- 's solution men tioned in the editor ~'s introduction. Visual insight in this case clearly indicates that it is impossible to manoeuvre the square in a position which produces a higher value for b 1 u.
" ' , .,
-
a
.
-' - " ' "
/
,
/
~~1 ~- )..-..-...~ ....: Fig. 3.
Largest flat square in cube
Selected Works of G. de Josselin de Jong
347
G. de JOSSELIN de lONG
212
4. CUBE IN TESSARACT
A similar procedure is now applied to the construction of a cube with edge length b, that can be inserted in a tessaract with edge length a, such that b >0. The tessaract is the body represented in Figure 4 by ABC D E F G HA' B' C D'E' P G' H'
The cube, that has to fit in it, is shown in Fig. 6 as P Q R S P' Q' R' S. Visual intuition suggests that the most favourable position for the cube is obtained, when its centre coincides with the centre of the tessaract. Let m and m' be the intersection points of the diagonals of the opposite faces P Q R S and P' Q' R' S' of the cube (Fig. 6). The centre M of the cube lies halfway mm'. For the tessaract its centre M lies halfway the centres n and n' of the two opposite faces B F F' B' and D H H'D' (Fig. 4). These faces are opposite, because, although being parallel, they are not in a common cubic boundary cell. In order to position the cube in the tessaract the choice is now made to superimpose the lines mMm' and nMn' so that the points M coincide. Further, the line nmMm'n' is considered to be the axis of rotation for a plane fJ, that will contain the diagonal plane PRR'P' of the cube. The plane fJ intersects the upper and lower faces of the tessaract (E'F'G'H' and ABC D in Fig. 4) in lines parallel to the diagonals in those faces. The segment R R' in face E' F' G' H' is shown separately in Fig. S. Its position is specified by the parameters A and p.. Its length b then satisfies the relation RR' = PP' = b = a h - 2"-
(4)
The symmetric positions of PP' and R R' in the upper and lower faces guarantee that P R R' P' is a rectangle. The lengths of the upright edges of this rectangle are b V2 and with Pythagoras' theorem it follows from Fig. 4 that (5)
(Note that the distance between the faces ABCD and E' F' G' H' is equal to a h.) Rotation of the plane fJ around the axis n m M m' n' varies the parameter p.. In Fig. 5 the situation P.,
G
'-f-..ll- ". If '
jD""' --H-~ !i.If"---I\-'-c--+i '... J= .····".."·I('· '.
!\\ ", ,
"'. "
'.
:\' '... :. " 1 ...
'.
\,
; ___ \
/""_~'_'
: \.
\,
'.;
\,
: \i\ .:
.;:
,,_ :.
c
\: \.
P
Teuaract inlCneClod by threo-spacc through ,", perpcndjcular 10 !be rotation LUI
l:" t~~1' 2~ 'l _. ,,
.
Q -...!'~/1
-
·
•• ••• I
;
.u
I
•
I---,m~''--l
(" -z~ )(z'
I
1 Fi"
I . Shape of inlerXction body willi plane IIIlrIml to PR
Selected Works of G. de Josselin de Jong
Fi" 9. Hal&""- limitin& lhc: si.l.c of SO in plane T through '" DOmIaI to PR
351
G. de lOSSELIN de lONG
216
6. RESULT
Solving (6) for A, after
1.1.
has been eliminated by means of (5), gives
,
~ = + aV2(b 2 - a 2 )1 / (bV2 + a). Only the upper sign is applicable, since A cannot be negative. Use of (4) then
gives
, (aV2 - bXbV2 - a) = aV2(b' - a')' .
(7)
There is only one root, viz.
(8)
b/a~I.OO7435 .
This result shows. that there exists indeed a cube with edges larger than those of the tessaract. From (4) and (5) it follows that
A/
a~O.2034
, "/ a~O.0864.
(9)
In the solution of the problem in the Am. Math. Monthly, mentioned in the editor prilRe-'s introduction, the ratio (8) was given as a root of an equation of degree eight. instead of the more simple equation (7). This was due to a less adequate elimination of>.. and p. from (4), (5) and (6). Fig. 10 shows how the cube fits into the tessaract. It may be noted that the point Q of the cube is located in the face BeG F of the boundary cell ABC D E F G H. Since both P and P' are situated in the face ABC D of that boundary ceU the entire face P Q Q' P' is located in that cell. Similarly. the face S R R' S' is a subset of the opposite boundary cell A' B' C D' E F' G' H'. Omy the two faces P Q Q' P' and S R R' S' of the cube are located in the boundary space of the tessaract. All other planes of the cube are in the interior. The vertices, however, are all situated on the boundary of the tessaract.
352
Soil Mechanics and Transport in Porous Media
217
CUBE IN TESSARACT
Fig. 10. Cube placed in the tessaract 7. COORDINATES
In orde r to facilitate verification of the resuh, the coordinates of the vertices of the cube ~re given bel~w as located in the tessaract of Section I with vertices (± fa'±"2a, ± fa' ± Ta). The lower signs in these expressions relate to the coordinates of the points R', S', P', (I. I
I
.~
_ I
_
I
I
. ,- _
1
P.R' = ( ± (Ta~"2(A+J..I)v2)' + "2a' + (Ta - "2(A - J..I)v2),+Ta)
Q, S' -- (+ .1a + ('!a _ 2AJ..I) - AV2) ' + .1a) - 2 ,- 2 a ' + (.1a 2 2 11
. t::I
_
11
_'-I
R,P' = (::t(Ta - "2(A - J..I)v2), ± Ta. +(Ta - T (A + J..I) v 2), ±"2a) S , Q' -- ( -+ (.1 2 a -,\ V2) . +- (.1 2 a _ 2AJ..I) a ' -+ .1 2 a' +.1) - 2a . It is possible 10 verify by linear algebra that the conditions guaranteeing P Q R S P' (I R' S' to be a cube, lead to equations (4), (5) and (6).
Selected Works of G. de Josselin de Jong
353
6 Complete Bibliography of Scientific Publications
6.1 List of publications The numbering of the papers is identical to the original numbering made by Gerard de Josselin de Jong, and therefore still completely in line with the actual hardcopy archive of the papers. [1] De spanni ngvcrdcJing rondom vertikale in zandige dekterreinen geboorde holten in stand gchoudcn door zwarc vloeistor. (T he stress distribution around a vertical hole ill oohesionless soil. ), G. de Josselin de Jong & J. Gcertscma, De IngeniclI.f, 1953, pp MI-M5
[2] Consolidation around pore-pressure meters, G. de Josselln de Jong, J. Appl. Phys ., Vol. 24, no. 7, 1953, pp 922-928 , American Ins titute of Physics [3J Wat scbeurt cr in de grond tijdens bet heien? (What happens in the soil during piled-driving?), Dc fngenicur, 1956, pp B77-B88 [4] Discussion, Proc. 3m Int. Conf. Soil Mech. fj Found. Eng_, Zurich, 1953, p 161
[5] L'cnt rainement de particules par Ie courant intcrsticicl, Proc. Symposia Darcy, Dijon, 1956, Publ. nr 41 of the UGGl, pp 139-1 47, UGGI
[6] Spoorbaanafsehniving (Slide in a. ra.i lway embankment) , LGM Mededelillgen, Vol. 1,1956, pp 7·40
[7] Verification of the usc of peak area for the quantitative differential thermal allaly~is, G. de Josselin de JOIIg, J. Am. Cerum. Soc., Vol. 40, no . 2, 1957, pp 42-49 , Ame ri can Cerl).mi c Societ y
354
6.1 Complete list of publications
355
[8] A capacitative cell apparatus, G. de Josselin de Jong & E.C.W.A. Geuze, Proc. ~th Int. Con/. Soil Mech. (1 Found. Eng., London, 1957, Vol. I , pp 52-55, Butterworth Scientific Publie"tiOlI~ {9] Application of stress functioilll to consolidation problcm~ , G. de Jossclin de Jong, Proc. ~th Int. Con/. Soil Mech. (1 Found. Eng., London, 1957, Vol. 1, pp 320-323 [10] Discussion, Proc. ~th Int. ConI- Soil Mech. fj Found. Eng. , London, 1957, Vol. 13, pp148-149
[11] Grafis 119-134 [19] Singularity distri bution~ for the analysis of multiple-fluid flow through porous media, G. de Jossclin de Jong, J. Geophys. Res. , Vol. 65, 110.11, 1960, pp 3739-3758, Americall Geophysical Unioll [20] Moire patterns of the membrane analogy for ground-water movement applied to multiple fluid flow, J. Geophys. Res., Vol. 66, no. 10, 1961, pp 3625-3628, American Geophysical Un ion [21) Discussion, Proc. Sthe Int. ConI. Soil Mech. fj Found. EfI9., Paris, 1961, Vol. 3, pp 326-327 21 a. Disc::u!i>lion of paper by J acob Bear Oil the Tensor Form of Dispersion in PoroW! Media, G. de Joslielin de Jong & M.J. Bosscll, J. Geophys. Res., Vol. 66, No. 10, 1961 , pp 3623-3624 21 b. Refraction moire analysis of curved surfaces, Proc. Symposium Shell Research, Delft, 1961, Nort h-Holland Publishin.g Comp., pp 302-308
Selected Works of G. de Josselin de Jong
355
356
6.1 Complete list 01 publications
[22] Vacuum voorgespannen zand als coustructiemateriaal , I (Vacuum prestressed sand as construction material, I), De IngenieuT, 1962, 8127-8134 [23) Vacuum voorgcspannen ?and als oonstructicmateriaal , II (Vacuum prestressed sand as construction material, 11) , De IngenieuT, 1963, 8113-8121
[24) Discussion, p=. EUT. Con/. Soil Mech. 1963, p 35 [25] Discussion, p=. EUT. Con/. Soil Mech. 1963, p 76
e e
Found. Eng. , Wiesbaden, Found. Eng., Wicsbaden,
[26] Consolidatie in drie dimensies, I, LOM Medelingen, 7, 1963, pp 57-73 [21] Consolidatie in drie dimensics, II, LOM Medelingen, 8, 1963, PI> 25-38
[28J Consolidatie in driO! dimO!nsies, 1II, LOM Medelingen, 8, 1964, PI> 53-68 [29] A many-valued hodograph in an interface problem, G. deJooselindeJong WateT ResouT. Res. , Vol. 1, no. 4, 1965, pp 543-555, American Geophys ical Union
[30] Primary and secondary consolidation of a. spherical sample, G. de JosseJin de J ong & A. Verruijt, Proc. 6th lnt Conf. Soil Mech. Montreal, 1965, Vol. 1, pp 254-258
e Found.
Eng. ,
[31) Lower bound collapse theorem and lack of normality of strainrate to yield surface for soils, G. de Josselin de Jong, Proc. l UTAM Symp. on Rheology and Soil Mechanics, edited by G. Kravtchen ko and P~d Siricys Grenoble, 1964, pp 69-78, Springer-Verlag [32) Discussion, G. de Josselin de Jong, Proc. EUT. Conf. Soil Mech. fj Found. Eng., Oslo, 1968, pp 199·200, Norwegia n Inst it ute of Technology [33] Consolidation models oonsisting of all assembly of visoous elements or a cavity channel network , G. de Josscliu de Jong, Geotechniquc, Vol. 18, 1968, Pi> 195-228 , Till.) Institu te of C ivil I~ ng i" ....cn;
[34] Generating functiOIl~ in the theory of flow through porous media, G. de Jossclinde Jong , Chapter 9, Flow Through Porous Media, Acadell1ic Press, 1969, pp 377-100 [35J Etude photo-clastique d 'un empilement de disqucs, G. de J085Ci in de J ong & A. Verrllijt, CahieT Groupe Fmw;ais de R/u;ologie, Vol. 2, No. 1, 1969, pp 73·86
(36J The double slidiug, free rotating model for granular assemblies, G. de .Jossclill de Jong,Giotechnique, 1971 , pp 155-163, The Im;l.itulioll of C ivi l Eligi llccl"s [371 Discussion, Proacding., Roscoc Memorial Symposium, Cambridge, 1971, pp 258-262 (38J Dispersion of a point injection in an anisotropic porOIlS med ium, Socorro Report, New MCJl:ioo 87801 , 1972 38 a. Dispersion in fissured rock, G. de Jossclin de Jong & Shao Chill Way, New Mexico 87801, 1972 [39J Dispersion described by differential CC]uation developed with Lagrangian Correlation Fullction~ , RappoTt Geotechniek Delft, 1973
356
Soil Mechanics Mechanics and and Transport Transport in Porous Porous Media Media Soil
6.1 Complete list of publications
357
[40] The tensor character of the dispersion coefficient in anisotropic poroll8 media, G. de Josselin de Jong , Pmc. JA HR Congress, Haifa Israel, 1972, pp 259-267 [41J Photoel{l.';tic verification of a mechanical model for the flow of a grannlar material, A. Drescher & G. de Jossclin de Jong, J. Meeh. Phl/s. So/ieU, 1972, Vol. 20, pp 337-351, Pergamon Press [42J A limit theorem for material with internal friction, Pmc. Symp. PIMtic· ity, Cambridge, 1973, pp 12-21 [43J Aelotropie van gestrnctureerde materialen, Cement XXVI, No.4, 1974, pp 166- 176 [44J Rowe's stress dilatancy relation based on friction, Geotechniql.le, 178, pp 527-534 [45J Groudmechauische aspecten van korrelstapelingen, Procestechniek, Jaargang 31, No. 12, 1976 45 a. Evaluation of principle strCliS difference in photo-elastic granular mooia by use of a compensator in pure bedding, &pporl. Laborolorium voor Geolechniek Delft, J une 1976 [46] Mathematical elaboration of the double sliding free rotating model, Archi· wun Mechr.miki Stosowr.mej, 29, No.4, Wars1.awa, 1977 46 a. Model for the behaviour of granular materials in progress ive flow, Contribution Conf. in Jab/ona, Poland, 1976, lecture notes [47J Review of vortex theory for multiple fluid flow , Delft Progre&8 Repor/, 2, 1977, pp 225-236 [48J Constitutive relations for the flow of a granular a.ssembly in the limit state of stress, Speciality Session 9, Pmc. Int Conf. Soil Mech . & Found. Eng., Tokyo, 1977, pp 87·95 [49} Lecture notes Int. Center for Mech. Sciences (CISM), Udine, 1974, Model for t he behaviour of granular material in progressive flow [501 Improvement of t he lowerbound solut ion for the vertical cut off in a oohesive, frictionless soil, G. deJosselin de Jong, GeQtechnique , T echnical notes June 1978, pp 197-201, T he Institution of Civil Enginccrs [51) Vortex theory for multi ple fluid in thre(! dimetl.'iions, C.deJosselinde Jong, Delft Progren Report, 4, 1979, pp 87-102 [52J Diskontinuitiiten in Grellzspannungsfelderu, Geotechniek, Jahrgang 2, Heft 1, 125-129 (translated by Smoltczyk) [53) Het verloop van een drukstoot door cen paal, K IVI Publicatie, Vreed.enburghdag 1977, 19 oktober 1977 [54} Application of the calcu l u~ of variations to the vertical cut off in cohesive frictionlCliS soil, G. de Jesselin de Jong, Geotechnique, 30, No.1, 1980, pp 1-16, The institutioll of Civil Enginccrs
Selected Works of G. de Josselin de Jong
357
358
[55)
6. 1 Complete list of publications
11I 11~traties (Illustrations), Uitgcgcven door de vakgrocp GL'OtL'chlliek tel" ge legenllcid \11.1l het afscheid van pTof.dr.ir. G. dc .TOSl:!din de J ong, June
25, 1980 [5 6 J 1'rilmte 1.0 Professor de Jossditl dc Jong, LGM McilcddiJlgcJI, Part XXI, No.2, June 1980 [57J A limit theorem for soils, Delft I'rogl'Css /lCf'O,·t, 5, 1980, PI) 280-291 [58J A variational fallacy, C. de JC>liscJ iu de Joug. Geott:C/wiqu c, JUlie H181 , Vol. :11 , lIu.2, pp 289·290. Tllf" rll~ti t.m i(>1) of Civil Enginf't'1'S [59 J The si multaneous flow of fresh and salt water ill aquifers of large horizontal extension determined by shear flow and \"Ol'tex 1heory, C. de J()Iisclin de .Iong, Proc. Euromech. Colleg., Edited by A. Verruijt & F. B. J. Baremls, Sept. 198 1, pp 75·82, A. A. Dalkema [6 0 J Behavior of an elasto-plastic doubk...sliding frL'(.... rotating material if subjectoo to an ideal sim ple shear test., P''OC. IUTAM Conf. on DeformatiQn alld Failure of Gm nu/Ilr Materials, Edited by Vermeer & Luger, Delft, 1982, P)I 555-562 [61 J Cube with edges larger tlmu lilcl!;e of the endo~i ug te;&'\ract, C. de .Josscl ill de J Ollg, Nieuw An;hicf voor de lViskuude, 4, Vol. :1, I!JM5, Pi> 20!J·2 17 (+ introduction: pp 207.208 ), Wiskulldig GCHootsdoap
[(2) 'l'riHlsversc d is pcr" io\1 from (1 \1 originally sharp frt.",h·salt intcrfao:.'e caused by shear tlow. G.de.los. 55-59, Elscvier
1631 EI'.,"_pI,","i' ""i,,, of "" "",hI, , 1i,Ii"S mod,I i" ,,,,d,,in,,I 'impI' shear tests. C.de JOli.'\Clin de .Jong, Geoleclmique, 38, :'l"o. 4, 1988 , pp 5:1:1-555 (+ 63a DisCIlS~ioll G. de Josseli ll de Jong Geoteclmique, 39, No.3, 1989, pp 565-5(6). T he Institution of Civil Engineers
[64] KC\'erling [luisman Lezing, 1989 [65 J Co-rota tio na l solution in simple shear tests. IYro/1! Memorial Symposium , Oxford, 1992. Proceedings Predictive Soil i....It.'
